GIT quotient conjecture for semistable syzygy points of canonical curves

Let $g$ and $p$ be integers, and let $\operatorname{Hilb}(\mathbb{P}^{g-1})_{p,2}^{\operatorname{ss}}$ denote the locus in the Hilbert scheme of genus $g$ canonically embedded curves whose $(p,2)$-syzygy point is semistable. Let $\lambda$ and $\delta$ be the standard Hodge and boundary divisor classes on $\overline{\mathcal{M}}_g$. Syzygy-GIT quotient conjecture. There is an isomorphism

$$\operatorname{Hilb}(\mathbb{P}^{g-1})_{p,2}^{\operatorname{ss}}/\\!/\operatorname{SL}(g) \simeq \operatorname{Proj} \bigoplus_{m\geq 0} \mathrm{H}^0\left(\overline{\mathcal{M}}_g, m\left(\left(8+\frac{4}{g}-\frac{(g-1)(g-2)}{g(g-p-1)}\right)\lambda-\delta\right) \right).$$

This would give a GIT construction related to the Hassett–Keel program, but the source presents it as perhaps overly optimistic and does not establish the claimed isomorphism.