Euler system existence and nonvanishing conjecture for rank-critical Panchishkin representations

Let $V$ be a $p$-adic Galois representation over a number field $K$, Hodge--Tate at the primes above $p$. For each prime $v$ above $p$, a Panchishkin subrepresentation is a $G_{K_v}$-stable subspace $V^+_v\subset V$ whose Hodge--Tate weights are all at least $1$, while those of $V/V^+_v$ are all at most $0$. Let $r(V)=\max(0,r_0(V))$, where

$$r_0(V)=d_{-}(V)-\sum_{v\mid p}\dim_L\operatorname{Fil}^0\mathbb{D}_{\mathrm{dR}}(K_v,V),$$

and call $V$ $r$-critical if $r(V)=r$ and $r(V^*(1))=0$. Say that $V$ satisfies the rank $r$ Panchishkin condition if it has Panchishkin subrepresentations $V^+_v$ for all $v\mid p$.

Euler system existence and nonvanishing conjecture. If $V$ is $r$-critical and satisfies the rank $r$ Panchishkin condition for $r\geq 0$, there exists a collection of cohomology classes

$$c_F\in\bigwedge^r H^1_f(F,V)$$

satisfying the Euler system compatibility relation, where $F$ varies over finite abelian extensions of $K$. Moreover, $c_K$ is non-zero if and only if the associated complex $L$-function of $V$ satisfies

$$L^{(r)}(V^*(1),0)\neq 0.$$

This predicts the existence of higher-rank Euler systems for representations satisfying the Panchishkin condition, together with a precise equivalence between nonvanishing of the base Euler-system class and the order-$r$ derivative of the associated $L$-function. The source does not provide resolution evidence, so the conjecture remains open.