Caruso–David–Mézard's genetic decomposition conjecture for deformation rings

Let $\overline{\rho}:G_K\rightarrow\operatorname{GL}_2(\mathbb{F})$ be irreducible and let $\tau$ be a tame inertial type of niveau $f$. The restriction $\overline{\rho}|_{I_K}$ defines a tame inertial $\mathbb{F}$-type, and hence the combinatorial gene $\mathbb{X}(\tau,\overline{\rho}|_{I_K})$ is defined. Write $\mathcal{O}$ for the coefficient ring and $R_{\overline{\rho}}^{\eta,\tau}$ for the associated deformation ring.

Genetic decomposition conjecture. There exists a decomposition

$$\mathbb{X}(\tau,\overline{\rho}|_{I_K})=\bigcup_{i=0}^{r}(\mathbb{X}_{j_i},\mathbb{X}_{j_{i+f}})_{j_i\leq j\leq j_{i+1}}$$

such that

$$R_{\overline{\rho}}^{\eta,\tau}\cong \widehat{\bigotimes}_{i=0}^{r}R_i,$$

where $R_i$ is a complete local Noetherian $\mathcal{O}$-algebra depending only on $(\mathbb{X}_{j_i},\mathbb{X}_{j_{i+f}})_{j_i\leq j\leq j_{i+1}}$.

This refines the preceding claim that the deformation ring is determined by the full gene, by asserting a tensor-product decomposition into local factors controlled by successive gene segments. The supplied text gives no resolution status.