small question about the vq-vae paper

hello!
thank you for your great work!

I have a question about loss function in paper.
L = log p(x|z(x)) + ||sg[z(x)] − e|| + β||z(x) − sg[e]||

the author mentioned that a third term exists because e can grow arbitrarily if it doesn't train as fast as the encoder parameters.
but I see that term only helps the encoder to be trained faster.

will it help the e to be trained faster too? but I assume that sg[e] is meaning that the e won't be trained by the term.
I hope this isn't a silly question ;) thx in advance.