The mod-$p$ compatibility conjecture for crystalline representations

Let $V$ be a crystalline representation of ${\rm G}_{{\mathbb Q}_p}$, let $T$ be a Galois-invariant lattice, and write $V\bmod p$ for the semisimplification of $T/pT$. Let $\Pi(V)$ be the associated unitary representation of ${\rm GL}_2({\mathbb Q}_p)$, let $\Pi^0(V)$ be a ${\rm GL}_2({\mathbb Q}_p)$-invariant unit ball, and let $\overline{\Pi}(V)$ be the semisimplification of $\Pi^0(V)/p$. The mod-$p$ compatibility conjecture. The representation $\Pi^0(V)/p$ has finite length, and $\overline{\Pi}(V)$ corresponds to $V\bmod p$ under the mod-$p$ correspondence described in the source. The conjecture would make $\overline{\Pi}(V)$ easier to compute than $V\bmod p$ from $D_{\rm cris}(V)$.