Why net E field inside (non ideal) wires doesn't end up to 0 ?

Question

I've come across an instructive paper which explains qualitatively what happens inside wires of a simple circuit during the transient state, leading to the steady state where there is an uniform E field everywhere inside the wires.

Here are two figures from the paper which are pretty much self explanatory :




What I don't understand is that sentence from figure 6 :

"pile-up of surface charge will continue until the net electric field does point to the left"

Why does the pile up of charges continues even after the net field becomes 0 ? Why charges keep piling up until the E field reverses ? Shouldn't the pile up stops when the net E field reaches 0 ?

Comments

Voting to close as high-school level. Might be more appropriate on PhysicsForums.com and the like.

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I first posted on Physics Forum, but no answer I understood. I just realized questions on this site seem to be much higher level indeed.