The Kisin-variety and gene conjecture for potentially crystalline deformation rings

Let $K/{\mathbb Q}_p$ be unramified, let $\overline\rho$ be absolutely irreducible, let $\lambda=(0,0)$ for each embedding, and let $t$ be a tame inertial type of level $f$. Let $\overline{\mathcal G\mathcal R}^{\lambda,t}_{\overline\rho}$ be the associated Kisin variety, equipped with its shape stratification, and let $\mathbb X$ be the gene attached to $(t,\overline\rho)$. Kisin-variety and gene conjecture.

1. The generic fibre of $R^{\lambda,t}_{\overline\rho}$ is determined by the Kisin variety equipped with its shape stratification.
2. The ring $R^{\lambda,t}_{\overline\rho}$ is determined by the gene.

This conjecture proposes that the geometry of the Kisin variety, together with its shape stratification, controls the generic deformation ring, while the combinatorial gene controls the entire deformation ring. The paper reports results determining the Kisin variety and its shape stratification from the gene, but the two asserted determination statements remain conjectural here.