Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time
There's an algorithm for multiplying an $N \times N^{0.172}$ matrix with an $N^{0.172} \times N$ matrix in $N^2 \operatorname{polylog}\left(N\right)$ arithmetic operations. The main identity used for it comes from Coppersmith's paper "Rapid multiplication of rectangular matrices", but the explanation for why it leads to $N^2 \operatorname{polylog}\left(N\right)$ instead of $N^{2 + \epsilon}$ is in the appendix of Williams' paper, "New algorithms and lower bounds for circuits with linear threshold gates".

This only works because Coppersmith's identity has some additional structure you can take advantage of, and the more recent MM algorithms don't seem to have this structure. That said, I'm not sure why one can't hope to extend this approach to $N \times N \times N$ matrix multiplication.