Degree-sum conjecture for singular ideals of symmetric-group Cherednik modules

Let $J_c(S_n,\mathfrak{h},\tau)$ be the ideal generated by singular vectors in the polynomial representation for $S_n$, with $\tau$ trivial. Assume that $p>n$, let $c\in\mathbf{F}_p$ be represented by an integer with $1\leq c\leq p-1$, and define

$$S=\left\{\frac{ap}{b}: 0\leq a\leq b<p\right\}$$

using the distinct rational values in this set. Suppose that no element of $S$ lies strictly between $c$ and $c+1$.

Degree-sum conjecture. The sum of the degrees in a minimal generating set of $J_{c+1}(S_n,\mathfrak{h},\tau)$ is exactly $n!$ greater than the corresponding sum for $J_c(S_n,\mathfrak{h},\tau)$.

The source reports verification for $n=3$ and in a small number of cases for $n=4$, leaving the general assertion open.