Universal conductor invariance of wild ramification for nearby cycles

Let $S$ be a strictly henselian trait with algebraically closed residue field, and let $X$, $x$, $\mathcal F$, and $\mathcal F'$ be as in Theorem: $X$ is a strictly local $S$-scheme essentially of finite type with closed point $x$ above the closed point of $S$, while $\mathcal F$ and $\mathcal F'$ are constructible complexes on $X_\eta$ with coefficients in finite fields of characteristics invertible on $S$. Assume that $\mathcal F$ and $\mathcal F'$ have universally the same conductors over $X$. Universal conductor invariance conjecture. The stalks $(R\psi\mathcal F)_x$ and $(R\psi\mathcal F')_x$ of the nearby cycle complexes have the same wild ramification, in the sense that for every $\sigma\in P_K$ their Brauer traces agree. This variant would extend the theorem that equality of wild ramification over $X$ is preserved by nearby cycles; whether universal equality of conductors suffices remains open.