Evans' Lex-Plus-Powers conjecture for graded Betti numbers

Let $S=\Bbbk[x_{1},\ldots,x_{n}]$, let $P=(x_{1}^{e_{1}},\ldots,x_{r}^{e_{r}})$, and let $F=(f_{1},\ldots,f_{r})$ be a homogeneous regular sequence with $\deg f_i=e_i$. Let $I$ be a homogeneous ideal containing $F$, and let $L$ be a lex ideal whose sum with $P$ has the same Hilbert function as $I$. Evans' Lex-Plus-Powers conjecture. For all $i$ and $j$,

$$b_{i,j}(L+P)\geq b_{i,j}(I).$$

This strengthens the Eisenbud–Green–Harris Hilbert-function assertion by imposing graded Betti-number inequalities. The conjecture is presented as open in the paper.