The characteristic-cycle pullback conjecture for log-clean ramification

Suppose that $X$ is purely of dimension $d$ and that $D$ has simple normal crossings. Let $I'\subset I$ be a subset containing $I_{\mathrm{T},\mathcal{F}}$, and let

$$D'=\bigcup_{i\in I'}D_i.$$

Assume that the ramification of $\mathcal{F}$ is $\log$-$D'$-clean along $D$, and that the inverse image

$$\tau_{D'}^{-1}\bigl(S_{D'}^{\log}(j_{!}\mathcal{F})\bigr)\subset T^*X$$

of the $\log$-$D'$-singular support $S_{D'}^{\log}(j_{!}\mathcal{F})\subset T^*X(\log D')$ by the canonical morphism $\tau_{D'}$ is of dimension $d$. Characteristic-cycle pullback conjecture. Then

$$CC(j_{!}\mathcal{F})=\tau_{D'}^{!}CC_{D'}^{\log}(j_{!}\mathcal{F})$$

in $Z_d\bigl(\tau_{D'}^{-1}(S_{D'}^{\log}(j_{!}\mathcal{F}))\bigr)$, where $\tau_{D'}^{!}$ is the indicated Gysin map. This conjecture predicts that, under the stated cleanliness and dimension hypotheses, the ordinary characteristic cycle is obtained from the logarithmic characteristic cycle by pullback; its status is not established by the supplied text.