The Lex Plus Powers conjecture on extremal graded Betti numbers

Let $R=k[x_1,\ldots,x_n]$, let $\mathbb A=\{a_1,\ldots,a_n\}$ be a non-decreasing list of integers with $a_1\geq 1$, and let $\mathcal H$ be a Hilbert function. Call $\mathcal H$ $\mathbb A$-lpp valid if there exists an $\mathbb A$-lex plus powers ideal $L$ with $H(R/L)=\mathcal H$. For ideals attaining $\mathcal H$ and containing an $\mathbb A$-regular sequence, write $\mathcal{LP}^{\mathcal H}_{\mathbb A}$ for the set of their Betti diagrams, ordered coefficientwise.

The Lex Plus Powers conjecture. If $\mathcal H$ is $\mathbb A$-lpp valid and $L_{\mathcal H,\mathbb A}$ is the $\mathbb A$-lex plus powers ideal attaining $\mathcal H$, then $\beta^{L_{\mathcal H,\mathbb A}}$ is the unique largest element of $\mathcal{LP}^{\mathcal H}_{\mathbb A}$.

Lex plus powers ideals are conjectured to have extremal Betti-number properties. The paper records this as an open conjecture and studies its relationship with the Hilbert-function and socle formulations.