Ellenberg–Jain–Venkatesh distribution conjecture for lambda-invariants

Fix a prime $p$. For imaginary quadratic fields $K$ in which $p$ is inert, let $\bb\b5_p(K)$ be the Iwasawa $\bb\b5$-invariant, and let $r$ be a nonnegative integer. Ellenberg–Jain–Venkatesh conjecture. The proportion of such fields for which $\bb\b5_p(K)=r$ is

$$p^{-r}\prod_{t>r}\left(1-p^{-t}\right).$$

This is a random-matrix heuristic for the distribution of Iwasawa invariants as the imaginary quadratic field varies. No resolution is given in the source.