Trace-classification conjecture for relative ind-coherent categories

Let $g:\mathcal Z\to\mathcal Y$ be a quasi-smooth map, let $\phi$ be an endomorphism of $\mathcal Y$ lifting to $\mathcal Z$, and let $\mathcal N$ be a conical singular-support condition preserved by $\phi$. Write $\mathcal Y^\phi$ and $\mathcal Z^\phi$ for the fixed-point loci, $g^\phi:\mathcal Z^\phi\to\mathcal Y^\phi$ for the induced map, and let $(-)_{\mathcal N^\phi}$ be the right-adjoint projection to $\operatorname{IndCoh}_{\mathcal N^\phi}(\mathcal Y^\phi)$. Trace-classification conjecture. Under the identification

$$\operatorname{Tr}_{\operatorname{DGCat}}(\phi_*,\operatorname{tIndCoh}_{\mathcal N}(\mathcal Y))\simeq \operatorname{IndCoh}_{\mathcal N^\phi}(\mathcal Y^\phi),$$

the class of $\operatorname{IndCoh}_{\mathcal N}(\mathcal Z\mid\mathcal Y)$ defined by the induced !-pullback corresponds to

$$((g^\phi)_*^{\operatorname{IndCoh}}(\omega_{\mathcal Z^\phi}))_{\mathcal N^\phi}.$$

This is presented as plausible and presumably not difficult, but is not established in the source.