Gasharov–Hibi–Peeva conjecture on Betti numbers over a power quotient

Let $S=\Bbbk[x_{1},\ldots,x_{n}]$, let $P=(x_{1}^{e_{1}},\ldots,x_{r}^{e_{r}})$, and set $R=S/P$. Let $I$ be a homogeneous ideal containing $P$, and let $L$ be the lex-plus-$P$ ideal with the same Hilbert function as $I$. Write $\bar I$ and $\bar L$ for their images in $R$. Gasharov–Hibi–Peeva conjecture. For all $i$ and $j$,

$$b_{i,j}^{R}(\bar L)\geq b_{i,j}^{R}(\bar I).$$

This is the analogue over $S/P$ of the Bigatti–Hulett–Pardue maximality theorem for lex ideals. The paper presents it as an open conjecture.