Stability conjecture for decompositions of Betti diagrams of powers of monomial ideals

Let $S$ be a polynomial ring and let $I$ be a monomial ideal in $S$ whose generators all have the same degree. For each $k$, let $\beta(S/I^k)$ denote the Betti diagram of $S/I^k$, and let a translation of a pure diagram $\pi_i(k)$ be a pure diagram whose nonzero shape is fixed while its length varies linearly with $k$. Let $P_k$ be the polytope of Betti diagram decompositions of $\beta(S/I^k)$, expressed in the coordinates of selected translated pure diagrams.

Stability conjecture. There is a $k_0$ such that, for all $k>k_0$, there are translations of pure diagrams $\pi_1(k),\ldots,\pi_m(k)$ such that every decomposition of $\beta(S/I^k)$ has the form

$$w_1\pi_1(k)+\cdots+w_m\pi_m(k).$$

Moreover, all $P_k$ have the same combinatorial type as a polytope $P_I$, and for every vertex $v$ of $P_I$ there is a function $h_v(k)\in\mathbb{R}^m$, rational in each coordinate, such that the corresponding vertex of $P_k$ is $h_v(k)$.

The conjecture proposes eventual stabilization of the combinatorial structure and rational variation of the vertices of decomposition polytopes for powers of an equigenerated monomial ideal. The preceding discussion recalls that pure-diagram decompositions exist by Boij–Söderberg theory; the asserted eventual behavior of these polytopes is the part posed as a conjecture, and no resolution is supplied here.