The inert-case valuation conjecture for $p$-adic regulator matrices

Let $p>3$ be a prime, and let $M_p^{\mathrm{inert}}$ be the measured space of matrices associated with the inert case. Define the random variable

$$X_f:M_p^{\mathrm{inert}}\longrightarrow\mathbb R_{\geq0},\qquad A\longmapsto v_p(\det(A)),$$

where $f=X^2+3Y^2$. Inert-case valuation conjecture. The random variable $X_f$ is distributed so that

$$v_p(\det(A))\ \text{ is in distribution equal to }\ X_f+2.$$

This is the conjectural inert-case computation of the valuation distribution; the source explicitly presents it as the remaining conjecture after the split and ramified cases are analyzed.