The Lex Plus Powers conjecture for the top socle degree

Let $R=k[x_1,\ldots,x_n]$, let $\mathbb A=\{a_1,\ldots,a_n\}$, and let $L$ be an $\mathbb A$-lex plus powers ideal with Hilbert function $\mathcal H=H(R/L)$. Let $\rho_{\mathcal H}$ denote the regularity of $\mathcal H$, and let $I\subset R$ contain an $\mathbb A$-regular sequence and attain $\mathcal H$.

Top-socle Lex Plus Powers conjecture. Then

$$\beta^L_{n,\rho_{\mathcal H}+n-1}\geq\beta^I_{n,\rho_{\mathcal H}+n-1}.$$

The paper states that proving this single-degree assertion is enough to establish the full socle conjecture, and therefore the Hilbert-function conjecture. Its status is not explicitly supplied beyond that equivalence, so it remains open here.