Greenberg–Benois conjecture on trivial zeros and $\ell$-invariants

Let $V$ be an absolutely irreducible $p$-adic Galois representation satisfying the Panchishkin condition, and suppose that Frobenius acts semisimply on its semistable module. Let $e_0$ and $t$ be the integers occurring in the decomposition of the local representation described in the source, let $r$ be the order of vanishing of the complex $L$-function $L(s,V)$ at $s=0$, and let $\ell(V^*(1))$ be the associated Greenberg $\ell$-invariant. Write $\mathcal{L}_p(S,V)$ for the $p$-adic $L$-function, $E_p(s,V)$ for its local Euler factor, $L_p(s,V)$ for the motivic Euler factor, and $L^{\operatorname{alg}}(s,V)$ for the complex $L$-function divided by its period. The notation $({E_p(s,V)L_p(s,V)^{-1}})^*$ means that all Euler factors vanishing at $s=0$ are removed.

Greenberg–Benois conjecture. If $\mathcal{L}_p(S,V)$ has $e_0+t$ trivial zeros, then $S^{e_0+t+r}$ divides it exactly and

$$\frac{\mathcal{L}_p(S,V)}{S^{e_0+t+r}} \equiv \ell(V^*(1))\left(E_p(0,V)L_p(0,V)^{-1}\right)^* \frac{L^{\operatorname{alg},(r)}(0,V)}{\log_p(1+p)^{e_0+t+r}(e_0+t+r)!} \pmod S.$$

This conjecture gives both the precise order of the trivial zero and the leading-term formula relating the $p$-adic $L$-function to the complex special value and the $\ell$-invariant. The supplied text gives no resolution status.