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

Let $p>3$ be a prime. Let $\mathbb Q_p^{(3)}$ be the unique unramified cubic extension of $\mathbb Q_p$, with ring of integers $\mathbb Z_p^{(3)}$ and cyclic Galois group generated by $\tau$. Define

$$M_p^{\mathrm{inert}}=\left\{\begin{pmatrix}1&1&1\a&\tau(a)&\tau^2(a)\tau(a)&\tau^2(a)&a\end{pmatrix}\mathrel{\Bigg|}a\in\mathbb Z_p^{(3)},\ a\ne0,\ \operatorname{Tr}_{\mathbb Q_p^{(3)}\mid\mathbb Q_p}(a)=0\right\}.$$

For $x\in\{\mathrm{split},\mathrm{ram},\mathrm{inert}\}$, let $M_p^x$ be the corresponding space of matrices, equipped with its canonical finite measure, and let $\mathcal K_p^x$ denote cyclic cubic fields with the corresponding splitting type at $p$. Uniformity conjecture. For every $x\in\{\mathrm{split},\mathrm{ram},\mathrm{inert}\}$ and $i\in\mathbb Z_{\geq0}$,

$$\lim_{D\to\infty}\frac{\#\{K\in\mathcal K_p^x\mid v_p(R_p(K))=i\text{ and }|d(K)|\leq D\}}{\#\{K\in\mathcal K_p^x\mid |d(K)|\leq D\}}=\operatorname{pr}(X=i),$$

where $X:M_p^x\to\mathbb R_{\geq0}$ is the random variable $X(A)=v_p(\det(A))$. The conjecture asserts that the matrices arising from cyclic cubic fields are distributed like random matrices in $M_p^x$; this is motivated by numerical observations and remains open, particularly in the inert case.