Local deep congruence conjecture for semistable representations

Let $\mathcal{L},\mathcal{L}'\in\mathbb{Q}_p$ be admissible, let $k$ be the weight, and let $C_{p,k}$ be the constant appearing in the conjecture. Let $V_{k,\mathcal{L},\varepsilon}$ denote the semistable representation associated with weight $k$, $\mathcal{L}$-invariant $\mathcal{L}$, and Atkin--Lehner sign $\varepsilon$. Local deep congruence conjecture. If

$$v_p(\mathcal{L}+\mathcal{L}')\ge -C_{p,k},$$

then $V_{k,\mathcal{L},1}$ and $V_{k,\mathcal{L}',-1}$ are congruent modulo $p^{-v_p(\mathcal{L})+1}$. This conjecture is the local representation-theoretic analogue of the global deep-congruence conjecture, and its general status is open.