Periodic cyclic homology conjecture for categorical DT theory of critical loci

Let $\mathfrak{M}$ be a quasi-smooth QCA derived stack, let $\mathcal{N}=t_0(\Omega_{\mathfrak{M}}[-1])$ be its associated d-critical locus, and let $\mathcal{N}_{\mathrm{st}}$ be the scheme obtained by removing the specified conical closed substack. Let $\phi_{\mathcal{N}}$ be the perverse sheaf of vanishing cycles with monodromy $T_{\mathcal{N}}$, constructed using the canonical orientation data, and let $\operatorname{vdim}\mathcal{M}$ be the rank of $\mathbb{L}_{\mathfrak{M}}|_{\mathcal{M}}$. Periodic cyclic homology conjecture. There should be an isomorphism of $\mathbb{Z}/2$-graded vector bundles on $\operatorname{Spf}\mathbb{C}((u))$ with connections

$$(\operatorname{HP}_{\ast}(\mathcal{DT}_{\mathrm{dg}}^{\mathbb{C}^{\ast}}(\mathcal{N}_{\mathrm{st}})),\nabla_u)^{\mathbb{Z}/2}\cong\left(H^{\ast+\operatorname{vdim}\mathcal{M}}(\mathcal{N}_{\mathrm{st}},\phi_{\mathcal{N}}|_{\mathcal{N}_{\mathrm{st}}})\otimes_{\mathbb{Q}}\mathbb{C}((u)),d+T_{\mathcal{N}}/u\right).$$

This is the precise local relation between categorical DT theory and cohomological DT theory, extending the preceding globalization conjecture; the source does not give a resolution.