The product decomposition conjecture for deformation spaces and stratified Kisin factors

Let $S$ be the set of indices at which the projection of the Kisin variety to the corresponding factor of $(\mathbb P^1_{k_E})^f$ is constant, and write $0=i_1<i_2<\cdots<i_r$ for its elements with $i_{r+1}=f$. Set $S_j=\{i_j,\ldots,i_{j+1}-1\}$, and let $\mathcal V_j\subset\prod_{i\in S_j}\mathbb P^1_{k_E}$ be the factor associated with the corresponding fragment of the gene. The stratification on the Kisin variety induces stratifications on the $\mathcal V_j$. The product decomposition conjecture. With this notation, the deformation space decomposes as

$$D^\psi({\bf v},{\rm t},\overline\rho)=D_1\times D_2\times\cdots\times D_r,$$

where each $D_i$ depends only on the stratified variety $\mathcal V_i$. This is presented as a refined version of the conjecture that the whole stratified Kisin variety determines the deformation space: it predicts that the deformation space decomposes according to the independent fragments of the gene.