Aside from what was said about the lead tubing and tank, as well as the pressure and the pump, look at another issue with the configuration in the picture.
The T is arranged A-B-A. The A-A path is straight through, large diameter and goes to the washing machine. When the washing machine is filling up on the A-A path, this will even cause under-pressure on the B lead to the coffee machine.
In these circumstances, I personally would have looked for an A-A-B T-piece, so as to create an A-B straight through path.
Another issue with the washing machine is water hammer. The washing machine's electrical inlet valve will open/close so fast that upon closing a water shock will be felt in the feeding tube. Question is if the coffee machine can handle that. I would anyhow introduce a dampening device to absorb the water hammer.
Still, given the lead/tank circumstances, there may be other priorities.

As to a Reversed Osmosis (RO) system, you need sufficient pressure and at least a little bit of flow to drip-by-drip filter clean water into a buffer tank and dirty water into the sewage. RO systems are always on and many of these will continue to let "dirty" water pass even when the buffer tank is filled. This buffer tank is pressurized and would potentially be able to feed the coffee machine. Question then is capacity of tank versus consumption by coffee machine and speed at which RO filter fills buffer tank. Faucets for pulling RO water usually do not provide huge flows and are small diameter, like the tubes involved.
For the RO to work, sufficient pressure is needed, either from the mains or an in-line pump.
My RO filter at home has an in-line valve that shuts off the mains water when the buffer tank is completely filled.

An issue with RO water is the mineral content is so low the water hardly conducts electricity any more and coffee machine boiler level detection may not properly function. Also, RO water can cause metal in the coffee machine to dissolve due to its extreme osmolality. Do not drink RO water; its pH is too low to be healthy and its low osmolality will demineralize you. Or, add salts before drinking and feeding to the coffee machine. I did not find proper kit/consumables in the consumer market to add the salts in the proper way and place in a RO system. With my Andreja Premium, I do not add salts and run some heavy metal risk. I do not notice the boiler water level issue. Proving how clean the RO water is, I can say the pour over tank does not build any dirt, algae, or whatever in years of usage.

Regards
Peter
Netherlands
Europe

An update on my low pressure adventures with the VII:

I've been using the machine now for just over a week without any problems. My plumbing set up is 3/8" JG piping to the machine via an in-line simple carbon filter. The flow isn't that great - 3L/min with the filter. And the pressure, I imagine is very low - the manometer needle on the left (for pump pressure) is just above the resting point!

However, despite this, the pump primed ok first time, and the boilers fill without a problem. After drawing off lots of water from the steam boiler, the pump kicks in to draw water, the needle flickers to 4bar, and drops to resting once the boiler fill is over. On brewing, I'm getting just under 9bar on the manometer. I've got a pressure PF and it's interesting seeing the pressure climb. It gets to 8bar without a problem, and then a 2-3 second delay before hitting the 9bar. Like a slow ramp up. I've read that rotary output is affected by input pressure...so even though it's achieving the desirable 9bar mark...do you think I should tweak the pump to deliver a slightly higher pressure than is already set?

The happy ending to all this is that this machine is fantastic! The plumb in nature of the setup is SO convenient - I don't think I could go back to hand-fill. The coffee it produces is consistently excellent. I had to work hard with the Silvia to get a good shot...with the VII, it's a lot more easier.

Thanks again for all your input guys! Cheers! Steve

Thank you for posting a followup and I'm glad that everything worked out OK.

If your pump is as pictured on the S1 forum website I referenced earlier, it is equipped with a balanced bypass valve and this compensates for changes in inlet pressure. EXACTLY how well it does this is yet to be reported but certainly the pump mfg feels it "does the job" as intended.

As regards the quick ramp to 8 bar and the 2-3 second delay to 9 bar, I attribute this to the balanced bypass feature of the pump but that is based on a foggy memory of operating a rotary pump machine for a month or so.

steem21 wrote:An update on my low pressure adventures with the VII:

I've been using the machine now for just over a week without any problems. My plumbing set up is 3/8" JG piping to the machine via an in-line simple carbon filter. The flow isn't that great - 3L/min with the filter. And the pressure, I imagine is very low - the manometer needle on the left (for pump pressure) is just above the resting point!

I wouldn't worry about the pressure / flow rate.
A 1/4 HP, Series 1 Procon pumps ~50 Gal / hr (~3 L /Min) at 9 bar according to the charts on their WEB site. That is the size of an external pump on a commercial unit. I doubt the pump in your machine can outrun your supply.

steem21 wrote: The coffee it produces is consistently excellent.

Why mess with a good thing?

Mark