Periodicity conjecture for higher a-numbers under the transformation r to (r+1)p+1

Let $p$ be the characteristic and let $a^r(X_n)$ denote the $r$th higher $a$-number in the given $\mathbf Z_p$-tower. For two non-negative integers $r_0,r_1$ satisfying

$$r_1=(r_0+1)p+1,$$

let the minimal period of the sequence $a^r(X_n)$ mean its least eventual period as a function of $n$.

Periodicity conjecture. The minimal periods of the sequences $a^{r_1}(X_n)$ and $a^{r_0}(X_n)$ are equal.

The conjecture concerns a structural relation between higher $a$-number formulas for paired values of $r$. The supplied context gives no resolution status, so it remains open in this record.