Let $X$ be an algebraic variety. Let $\mathcal I_{\Delta}\subset\mathcal O_{X\times_kX}$ be the ideal sheaf defining the diagonal $\Delta\subset X\times_kX$. Regard $\mathcal O_{X\times_kX}/\mathcal I^{n+1}$ and $\mathcal I/\mathcal I^{n+1}$ as $\mathcal O_X$-modules through the first projection.

The question is: Are these $\mathcal O_X$-modules locally free of constant rank over some open dense set of $X$? When $n=1$ is well known, what about for n>1?