Goss–Gekeler slope multiplicity congruence conjecture

Let $q$ be the cardinality of the coefficient field, let $\alpha$ be a slope, and let $d(k,\alpha)$ denote its multiplicity for weight $k$. For integers $k_1,k_2$ and some integer $n\geq\alpha$, assume

$$k_1,k_2>2\alpha,\qquad k_1\equiv k_2\pmod{(q-1)q^{n-1}}.$$

Slope multiplicity conjecture. Under these hypotheses,

$$d(k_1,\alpha)=d(k_2,\alpha).$$

The conjecture is based on computational data and was previously formulated in the cited work. Hattori proved a related stability result under different hypotheses, but the stated congruence conjecture is not resolved in the source.