Booher–Cais conjecture for higher a-numbers in minimal-break-ratio towers

Let $\{X_n\}$ be a $\mathbf Z_p$-tower of curves totally ramified over one point of $X_0\simeq\mathbf P^1$, with minimal break ratios and ramification invariant $d$. For a positive integer $r$, define

$$a^r(X_n)=\dim_k\ker\left(V_{X_n}^r:\operatorname{H}^0(X_n,\Omega^1_{X_n})\to\operatorname{H}^0(X_n,\Omega^1_{X_n})\right).$$

Booher–Cais conjecture. For each $r\geq 1$, there exist an integer $m_r$ and functions $b_r,\nu_r,\lambda_r:\mathbf Z/m_r\mathbf Z\to\mathbf Q$ such that, for $n\gg0$,

$$a^r(X_n)=d\cdot\frac{r(p-1)}{2(p+1)((p-1)r+(p+1))}\cdot p^{2n}+b_r(n)\cdot p^n+\lambda_r(n)\cdot n+\nu_r(n).$$

This is the specialization of the broader regularity conjecture to towers over $\mathbf P^1$ with minimal break ratios. The paper states that the formula is proved for the specific class under study, but the supplied status is unknown; the conjecture is therefore recorded as open.