Bergdall–Pollack equidistribution conjecture for slopes of $\mathcal{L}$-invariants

Let $p$ be a prime number, let $N$ be an integer relatively prime to $p$, and for each $p$-newform $f\bin S_k(\Gamma_0(Np))$ let $\mathcal{L}_f$ be its $\mathcal{L}$-invariant. Normalize the $p$-adic valuation by $v_p(p)=1$, and define

$$Y_T:=\left\{\frac{2(p+1)}{(p-1)k}\,v_p(\mathcal{L}_f^{-1}): f\text{ is a }p\text{-newform in }S_k(\Gamma_0(Np)),\ k\leq T\right\}.$$

Bergdall–Pollack equidistribution conjecture. As $T$ tends to infinity, the set $Y_T\subseteq(-\infty,+\infty)$ is evenly distributed on $[0,1]$.

The paper confirms specific instances of this conjecture under local assumptions on the associated residual Galois representation, but the general statement remains open.