Has there been any progress in tightening the exponent in the result that polylog independence fools $AC_0$?
In his CCC'17 paper [1], Avishay Tal improved the bound to $$
\left(\log\frac{m}{\varepsilon}\right)^{O(d)}\,. \tag{1}
$$
You may want to check p.15:4 for a discussion. It also refers to (see Footnote 30 to a paper of Harsha and Srinivasan, which improves on (1)) and answers Tal's conjecture: $k$-wise independent, for
$$
k = \left(\log m \right)^{O(d)}\cdot\log\frac{1}{\varepsilon}\,. \tag{2}
$$
suffices to $\varepsilon$-fool size-$m$ depth-$d$ AC0 circuits.



[1] Tight Bounds on the Fourier Spectrum of $\mathsf{AC}_0$, A. Tal. CCC'17.

[2] On polynomial approximations to $\mathsf{AC}_0$, P. Harsha and S. Srinivasan. RANDOM 2016,