Socle lower bound conjecture for homology manifolds

Let $\Delta$ be a connected orientable homology $(d-1)$-manifold without boundary whose vertex links have the weak Lefschetz property. Let $\Theta$ and $w$ be as in Theorem 5.4, set $R=S/(I_\Delta+(\Theta))$ and $R'=R/wR$, and write $\operatorname{Soc}R'$ for the socle of $R'$. Socle lower bound conjecture. For $r\leq\frac d2$,

$$\dim_{\mathbf{k}}(\operatorname{Soc}R')_r\geq\binom{d+1}{r}\beta_{r-1}(\Delta).$$

This is posed as one of the paper's closing questions; no resolution is given.