Griffiths Electrodynamics example 12.13

"Consider a rectangular loop of wire carrying a steady current I . Picture the current as a stream of noninteracting positive charges that move freely within the wire. When a uniform electric field E is applied, the charges accelerate in the left segment and decelerate in the right one."

Afterwards he says:

"The current (I = λu) is the same in all four segments (or else charge
would be piling up somewhere)"

I'm assuming he is referring to the current after the influence of the electric field. Is it obvious that the current remains constant after the influence of the electric field? Why is it so? What does the "piling up" mean?

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  [1]: https://i.sstatic.net/YbHiT.png