Low-codimension perfect-pairing conjecture for compact type moduli

Let $\mathsf{R}^i(\mathcal{M}_{g,n}^{\operatorname{ct}})$ denote the codimension-$i$ part of the tautological ring of the moduli space of stable curves of compact type, and consider the pairing

$$\mathsf{R}^i(\mathcal{M}_{g,n}^{\operatorname{ct}})\times \mathsf{R}^{2g-3+n-i}(\mathcal{M}_{g,n}^{\operatorname{ct}})\longrightarrow \mathsf{R}^{2g-3+n}(\mathcal{M}_{g,n}^{\operatorname{ct}}).$$

Low-codimension pairing conjecture. For $i\leq 3$, these pairings are perfect. If $i=4$, the pairing has rank $\dim \mathsf{R}^4(\mathcal{M}_{g,n}^{\operatorname{ct}})$. The conjecture implies the analogous statement in cohomology; the case $i=0$ follows from the one-dimensional socle, and the source proves the case $i=1$, leaving the remaining cases open.