Finite-depth conjecture for étale sheaves

Let $X$ be a smooth variety over an algebraically closed field $k$ of characteristic $p\neq 2$, and let $\mathcal{F}\in D(X)$. Write $SS\mathcal{F}$ for its singular support and let $(x,\xi)$ be a smooth point of $SS\mathcal{F}$. The finite-depth conjecture. The complex $\mathcal{F}$ has finite depth at every smooth point of $SS\mathcal{F}$. The conjecture extends the paper's finite-depth theorem beyond the hypotheses treated there; the condition $p\neq 2$ ensures that there are enough test functions for the depth to be defined. Its general validity remains open.