The monoid conjecture for stratified embedded Kisin varieties

Let $\text{SEKV}$ be the monoid of shape-stratified closed subvarieties of $(\mathbb{P}^1)^n$, for varying $n$, with multiplication given by products. For a tame inertial type ${\rm t}$ and an absolutely irreducible representation $\overline\rho:G_K\to\operatorname{GL}_2(k_E)$, let $\overline{\mathcal{G}\mathcal{R}}({\rm t},\overline\rho)$ denote the associated stratified embedded Kisin variety, and let $R^{\psi}({\rm t},\overline\rho)$ be the deformation ring. Monoid conjecture. There exists a monoid morphism

$$R:\text{SEKV}\longrightarrow\{\text{complete noetherian }\mathcal O_E\text{-algebras}\},$$

where multiplication in the codomain is completed tensor product, such that

$$R^{\psi}({\rm t},\overline\rho)\simeq R\big(\overline{\mathcal{G}\mathcal{R}}({\rm t},\overline\rho)\big)$$

for all such $\overline\rho$ and all nondegenerate tame inertial types. This conjecture asserts that the deformation ring is determined functorially by the stratified embedded Kisin variety; the source gives no resolution.