This is incorrect and I think the crux of your misunderstanding of the processes.
When you bring two different gasses into contact, it's their relative concentration, aka concentration gradient, that drives diffusion. Concentration simply means the number of molecules of each substance per given area, Differences in pressure only matter to diffusion because, for gases, higher pressure means a greater difference in concentration.
It may help to bring this back to first principles. Unlike solids, gas molecules are constantly zipping around in random directions. If you've got only x molecules in one spot and y molecules in another, then from the first zip they are going to start intermingling. With each subsequent zip, around half the molecules will further move away from their initial point of concentration. It doesn't take that many zips before the concentration differences have disappeared. We call this equilibrium, but it's a mistake to think of it as static. The molecules continue to zip around, causing variations in concentration in any particular tiny volume.
I think I should link this here:
Fick's laws of diffusion - Wikipedia
Of course, if you put some sort of barrier between the two gasses, this process will slow. But it won't stop if there is any possible movement across the barrier. In your case of the gas tanks connected by a hose (of reasonable length), the tanks will reach equilibrium within several hours, maybe even a day, but certainly not "forever".