I came across a question today about vapour pressure, it was as follows

2 flasks X and Y, each containing liquid water, joined by a tube connecting their mouths (no vapour could escape) having a valve in the middle were immersed in water baths of 300K and 350K respectively, which were used to ensure that the flasks are maintained at a constant temperature. It is given that the vapour pressure of water is 22 torr at 300K and 40 torr at 350 K, and that the valve was initially closed. Now, the valve is opened, comment on the final pressure in both the flasks.

I was able to deduce that some amount of water would evaporate from Y and enter X, and this would continue until pressure in both the flasks is the same, however, I am not able to visualise the steady state of the system, and am confused on whether or not the vapour pressures can ever be the same, seeing vapour pressure is only dependent on temperature, and the temperatures of the 2 flasks will always be different.

So, what would happen after the valve is closed, and after the steady state is attained?