The tensor-induction local-global compatibility conjecture

Let $K=F_v$ be unramified, let $\pi$ be an automorphic admissible smooth representation of ${\rm GL}_2(K)$ over $\mathbb{F}$, and let $\overline{r}_v$ be its corresponding local Galois representation. Let $D_A(\pi)$ be the associated étale $(\varphi,\mathcal{O}_K^\times)$-module over $A$, and let $D_A^\otimes(\overline{r}_v(1))$ be the module attached to the tensor induction of the twist $\overline{r}_v(1)$. The tensor-induction compatibility conjecture. There is an isomorphism of étale $(\varphi,\mathcal{O}_K^\times)$-modules over $A$,

$$D_A(\pi)\cong D_A^\otimes(\overline{r}_v(1)).$$

This is the paper's proposed local-global compatibility statement; the surrounding theorem proves it under semisimplicity and sufficient genericity assumptions, while the conjecture removes those assumptions.