Pixton's completeness conjecture for the 3-spin relations

Let $\mathsf{FZ}^*(\overline{\mathcal{M}}_{g,n})$ be the space of $3$-spin relations on the moduli space of stable curves, and let $\mathsf{FZ}^*(\mathcal{M}_{g,n}^{\operatorname{ct}})$ be its restriction to the compact type locus. Define

$$\mathsf{R}^*_{\mathsf{FZ}}(\overline{\mathcal{M}}_{g,n})=\mathsf{S}^*(\overline{\mathcal{M}}_{g,n})/\mathsf{FZ}^*(\overline{\mathcal{M}}_{g,n}),\qquad \mathsf{R}^*_{\mathsf{FZ}}(\mathcal{M}_{g,n}^{\operatorname{ct}})=\mathsf{S}^*(\mathcal{M}_{g,n}^{\operatorname{ct}})/\mathsf{FZ}^*(\mathcal{M}_{g,n}^{\operatorname{ct}}).$$

Pixton's conjecture. The $3$-spin relations are complete in Chow and cohomology:

$$\mathsf{R}^*_{\mathsf{FZ}}(\overline{\mathcal{M}}_{g,n})=\mathsf{R}^*(\overline{\mathcal{M}}_{g,n})=\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$$

and

$$\mathsf{R}^*_{\mathsf{FZ}}(\mathcal{M}_{g,n}^{\operatorname{ct}})=\mathsf{R}^*(\mathcal{M}_{g,n}^{\operatorname{ct}})=\mathsf{RH}^*(\mathcal{M}_{g,n}^{\operatorname{ct}}).$$

The source states that the known relations are contained in the span of the $3$-spin relations and that this conjecture remains open; the paper proves new cases, including $\mathcal{M}_{6}^{\operatorname{ct}}$, $\mathcal{M}_{5,2}^{\operatorname{ct}}$, and $\mathcal{M}_{7}^{\operatorname{ct}}$.