The dimensions conjecture for fixed subspaces of the Cartier operator

Let $R\in\mathbb{F}_q[z]$ have degree at least $1$ and not be divisible by $z$, and write

$$R=cR_1^{e_1}\cdots R_k^{e_k}$$

for its factorization into irreducibles. Define $\lambda_0(S)=\Lambda_0(SR^{q-1})$. For a positive integer $m$, consider the vector space of polynomials $S\in\mathbb{F}_q[z]$ satisfying $\deg S\leq\deg R$ and $\lambda_0^m(S)=S$.

The dimensions conjecture. For every divisor $m$ of $\operatorname{lcm}(\deg R_1,\dots,\deg R_k)$, this vector space has dimension

$$1+\sum_{i=1}^k\gcd(m,\deg R_i).$$

In particular, the predicted dimension is independent of the exponents $e_i$. The conjecture would determine the period bound for the corresponding orbit under $\lambda_0$, but its status is not resolved in the source.