The $p$-primary structure conjecture for cuspidal divisor class groups of $X_1(p^n)$

Let $p$ be a prime and $n$ a positive integer. For the cuspidal divisor class group $\mathscr C_1^\infty(p^n)$, let $[p]$ denote multiplication by $p$:

$$[p]:\mathscr C_1^\infty(p^n)\to\mathscr C_1^\infty(p^n).$$

Define its $p$-rank to be the integer $k$ such that the kernel of $[p]$ has $p^k$ elements. The $p$-primary structure conjecture. If $p$ is a regular prime and $p^n\geq 8$ with $n\geq 2$, then the $p$-rank is

$$\frac12(p-1)p^{n-2}-1.$$

More precisely, under the same hypotheses, the number of copies of $\mathbb Z/p^{2k}\mathbb Z$ in the primary decomposition of $\mathscr C_1^\infty(p^n)$ is

$$\begin{cases} \frac12(p-1)^2p^{n-k-2}-1, & p=2\text{ and }k\leq n-3, \\ \frac12(p-1)^2p^{n-k-2}-1, & p\geq 3\text{ and }k\leq n-2, \\ \frac12(p-5), & p\geq 5\text{ and }k=n-1, \\ 0, & \text{otherwise}, \end{cases}$$

and the number of copies of $\mathbb Z/p^{2k-1}\mathbb Z$ is

$$\begin{cases} 1, & p=2\text{ and }k\leq n-3, \\ 1, & p=3\text{ and }k\leq n-2, \\ 1, & p\geq 5\text{ and }k\leq n-1, \\ 0, & \text{otherwise}. \end{cases}$$

The conjecture is based on limited numerical computations of the groups for prime powers below $800$ and concerns the unresolved structure of their $p$-primary parts.