Monomial-ideal reduction form of the Lex-Plus-Powers conjecture

Let $S=\Bbbk[x_{1},\ldots,x_{n}]$, let $(f_{1},\ldots,f_{r})$ be a regular sequence with $\deg f_i=e_i$, and set

$$P=(x_{1}^{e_{1}},\ldots,x_{r}^{e_{r}}).$$

Let $I$ be the homogeneous ideal containing this regular sequence. Monomial reduction form of the Lex-Plus-Powers conjecture. There exists a monomial ideal $M$ containing $P$ such that $I$ and $M$ have the same Hilbert function; moreover, $M$ may be chosen so that

$$b_{i,j}(M)\geq b_{i,j}(I)$$

for all $i,j$. The paper says this statement is equivalent to the Lex-Plus-Powers conjecture and that it is open; without the Betti-number condition, it has long been known to imply the Eisenbud–Green–Harris conjecture.