Milnor-type formula for the non-acyclicity class

Let $k$ be a perfect field, let $X$ be a smooth scheme over $k$, let $\mathcal F\in D_{\rm ctf}(X,\Lambda)$, and let $SS(\mathcal F)$ be its singular support. Let $Z\subseteq X$ be a closed subscheme and let $f:X\to Y={\rm Spec}k[t]$ be a morphism that is $SS(\mathcal F)$-transversal outside $Z$. The refined Gysin pull-back of the characteristic cycle by the section $f^*dt:X\to T^*X$ is

$$cc^Z_{X/Y/k}(\mathcal F):=(f^*dt)^!(CC(\mathcal F)),$$

a zero-cycle class on $Z$. Milnor-type conjecture. One has

$$\widetilde{C}^Z_{X/Y/k}(\mathcal F)=\widetilde{{\rm cl}}(\cc^Z_{X/Y/k}(\mathcal F))\quad\text{in}\quad H^0_Z(X,\mathcal K_{X/Y/k}),$$

where $\widetilde{{\rm cl}}$ is the composition of the cycle class map with the natural map to $H^0_Z(X,\mathcal K_{X/Y/k})$. This is the expected characteristic-cycle formula for non-isolated singular or characteristic points; the supplied text does not establish its resolution status.