Leal's ramification bound conjecture for nearby cohomology

Let $\mathcal S$ be a henselian trait with perfect residue field of characteristic $p>0$, with closed point $s$, generic point $\eta$, and geometric generic point $\overline\eta$. Let $(\mathcal X,\mathcal Z)$ be a semi-stable pair over $\mathcal S$, let $f:\mathcal X\to\mathcal S$ be proper, and let $\mathcal U=\mathcal X-\mathcal Z$. Let $\mathscr F$ be a locally constant and constructible sheaf of $\Lambda$-modules on $\mathcal U$ whose ramification at generic points of the horizontal part $\mathcal Z_f$ is tame. Let $\operatorname{lc}(\mathscr F)$ be the maximum of its logarithmic conductors at generic points of the special fiber $\mathcal X_s$. Leal's conjecture. The upper-numbering ramification of

$$R\Gamma_c(\mathcal U_{\overline\eta},\mathscr F|_{\mathcal U_{\overline\eta}})$$

is bounded by $\operatorname{lc}(\mathscr F)$. Thus the compactly supported étale cohomology of the generic fiber should have no upper-numbering slopes exceeding the largest logarithmic conductor along the special fiber. This is the ramification-bound formulation attributed to Leal and concerns the control of wild monodromy in étale cohomology; the source gives no resolution of the conjecture.