Gilmer's cut-number conjecture

Let $p\geq 5$ be a prime and let $q_p=(p-3)/2$. For a closed, connected $3$-manifold $M$, let $\operatorname{cut}(M)$ denote its cut number and let ${\mathfrak o}_p(M)$ denote the associated $p$-modular TQFT invariant.

Gilmer's cut-number conjecture. For every closed, connected $3$-manifold $M$,

$$\operatorname{cut}(M)\leq \frac{1}{q_p}{\mathfrak o}_p(M).$$

This conjecture is presented as a consequence of the preceding half-projective TQFT conjecture and is attributed in the source to Gilmer. Its resolution is not supplied in the source.