The distributional conjecture for normalized L-invariants

Let $N$ be an integer coprime to $p$, set $\Gamma_0=\Gamma_0(Np)$, and let $\overline{\rho}:\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)\to\operatorname{GL}_2(\overline{\mathbb F}_p)$ be modular of level $\Gamma_0(N)$. Let $S_k(\Gamma_0)^{\pm}$ be the subspace of $p$-new cuspforms on which $a_p=\pm p^{k/2-1}$. For each sign, define

$$\mathbf y_T^{\pm}(\overline{\rho})=\left\{\frac{2(p+1)}{k(p-1)}v_p(\mathcal L_f)\ \middle|\ f\in S_k(\Gamma_0)^{\pm}\text{ is an eigenform of weight }k\le T,\ \overline{\rho}_f\simeq\overline{\rho}\right\}.$$

Distributional conjecture. For each sign $\pm$, the sets $\mathbf y_T^{\pm}(\overline{\rho})$ become equidistributed for Lebesgue measure on $[-1,0]$ as $T\to\infty$. This conjecture proposes a statistical law for $p$-adic valuations of $\mathcal L$-invariants and is presented as a close relative of Gouvêa's slope distribution conjecture. The paper supplies numerical and heuristic evidence but no proof of the general statement.