How to synchronize a 2d cellular automaton in $\Theta(\sqrt{n} \log n)$ steps
A naive solution could be that each general first launch a binary counter in its row and counts the current time until $k\log(k)$ steps and then run the FSSP. To do this you just need to detect the value of $k$ and compute $k\log(k)$ in less than $k\log(k)$ steps: sending a signal back and forth toward the other end while incrementing a binary counter gives you k written in binary in roughly $2k+\log(k)$ steps, so you also know $\log(k)$ (length of the counter). Within the $\Theta(k\log(k))$ steps left you have much more time than necessary to compute the product $k\times \log(k)$. And you're done.

I'm pretty sure that (1) there are more elegant solutions, (2) there is a general acceleration/slowdown theorem that could also be used here.