Lex-plus-powers conjecture for graded Betti numbers

Let $R=K[x_1,\ldots,x_n]$, and let $I$ be a homogeneous ideal containing a regular sequence of degrees $2\leq a_1\leq\ldots\leq a_n$. Suppose that there exists a lex-plus-powers ideal $L$ with the same Hilbert function as $I$. Write $\beta_{i,j}(R/I)=\dim_K(\operatorname{Tor}^R_i(R/I,K))_j$ for the graded Betti numbers. Lex-plus-powers conjecture. For all $i$ and $j$,

$$\beta_{i,j}(R/I)\leq\beta_{i,j}(R/L).$$

This conjecture strengthens the Hilbert-function comparison by predicting extremality for every graded Betti number. The source describes substantial special cases, while the general statement remains open.