Radicality conjecture for the Hasse–Witt ideal of genus-4 double covers

Let $K$ be the coefficient field, let $C$ be a genus-$4$ double cover of an elliptic curve with defining parameters $a_0,\ldots,a_5$, and let $f_2$ be the polynomial defining the nonsingularity condition. Set

$$I_M\subset K[a_0,\ldots,a_5,1/f_2]$$

to be the ideal generated by the entries of the Hasse–Witt matrix of $C$.

Radicality conjecture. The ideal $I_M$ is radical for every prime $p\geq 5$.

This conjecture arose from computations involving superspecial genus-$4$ double covers of elliptic curves, particularly those with a unique degree-two morphism to an elliptic curve. Its status is not resolved in the supplied source context.