The Lex Plus Powers conjecture for Hilbert functions

Let $R=k[x_1,\ldots,x_n]$, let $\mathbb A=\{a_1,\ldots,a_n\}$, and let $I\subset R$ contain an $\mathbb A$-regular sequence. Suppose there exists an $\mathbb A$-lex plus powers ideal $L$ such that $H(R/I,d)=H(R/L,d)$. Let $\langle L_d\rangle$ denote the ideal generated by $x_1^{a_1},\ldots,x_n^{a_n}$ and the degree-$d$ forms in $L$.

The Lex Plus Powers conjecture for Hilbert functions. Then

$$H(R/\langle L_d\rangle,d+1)\geq H(R/I,d+1).$$

This is the Hilbert-function-growth formulation attributed in the source to Eisenbud, Green, and Harris. It is stated as open and is described as weaker than the full Lex Plus Powers conjecture.