The tensor-product conjecture for deformation rings of stratified Kisin varieties

Let $\mathbb{X}$ be a fixed nondegenerate gene, and write its stratified embedded Kisin variety as

$$\overline{\mathcal{G}\mathcal{R}}^{\mathrm{s}}(\mathbb{X})=\mathcal V^{\mathrm{s}}_0\times\cdots\times\mathcal V^{\mathrm{s}}_{r-1}$$

inside $(\mathbb{P}^1)^f$. Let $R^{\psi}({\rm t},\overline\rho)$ and $D^{\psi}({\rm t},\overline\rho)$ denote the potentially Barsotti–Tate deformation ring and its generic fibre. Tensor-product conjecture. The following hold:

$$R^{\psi}({\rm t},\overline\rho)\simeq R(\mathcal V^{\mathrm{s}}_0)\mathbin{\widehat\otimes}_{\mathcal O_E}\cdots\mathbin{\widehat\otimes}_{\mathcal O_E}R(\mathcal V^{\mathrm{s}}_{r-1}),$$

where each $R(\mathcal V^{\mathrm{s}}_j)$ is a local noetherian complete $\mathcal O_E$-algebra depending only on $\mathcal V^{\mathrm{s}}_j$. 2. $R^{\psi}({\rm t},\overline\rho)$ is the ring of power-bounded functions on $D^{\psi}({\rm t},\overline\rho)$ if there is no index $i$ with $X_i=X_{i+f}=\operatorname{Dom}_{i+1}(\mathbb{X})$.

The conjecture is a weaker integral refinement of the proposed product decomposition of the generic fibre. The source gives no resolution.