Relative affinization conjecture for the filtered nodal curve

Let $R$ be a commutative ring, and let

$$[\mathbb{A}^1/\mathbb{G}_{m}]\to \mathcal{M}_{cub}$$

be the map classifying the filtered nodal curve over $R$. Denote the corresponding filtered nodal curve by $\mathcal{N}_{\mathrm{fil}}$, and let $S^1_{\mathrm{fil}}$ be the filtered circle of Moulinos, Robalo, and Toën. Relative affinization conjecture. The relative affinization of $\mathcal{N}_{\mathrm{fil}}$ is equivalent to the filtered circle:

$$\operatorname{Aff}_{[\mathbb{A}^1/\mathbb{G}_{m}]}(\mathcal{N}_{\mathrm{fil}})\simeq S^1_{\mathrm{fil}}.$$

Consequently, for every affine scheme $X$ over $R$, there is an equivalence

$$\mathrm{HH}_*^{\mathrm{fil}}(X)\simeq \mathcal{O}\!\left(\operatorname{Map}_{[\mathbb{A}^1/\mathbb{G}_{m}]}\left(\operatorname{Aff}_{[\mathbb{A}^1/\mathbb{G}_{m}]}(\mathcal{N}_{\mathrm{fil}}),X\times[\mathbb{A}^1/\mathbb{G}_{m}]\right)\right).$$

This would explain the parallelism between tmf-Hochschild homology and filtered Hochschild homology through the degeneration from the nodal cubic to the cuspidal cubic. The source presents the assertion as a conjectural description.