Stabilization-index formula for square-free monomial ideals

Let $I_G=(m_0,m_1,\ldots,m_k)\subseteq \Bbbk[x_1,\ldots,x_N]$ be a square-free monomial ideal with

$$\bigcup_i \operatorname{supp}(m_i)=\{x_1,\ldots,x_N\}.$$

For a homogeneous ideal $I$ equigenerated in degree $r$, let $\operatorname{Stab}(I)$ be the smallest integer $D$ such that, for all $d\geq D$, the nonzero graded Betti numbers of $I^d$ agree with those of $I^D$ after the shift by $r(d-D)$. Stabilization-index conjecture.

$$\operatorname{Stab}(I_G)=\min\left\{n:\text{ there exists an }\mathbf{m}\in I_G^n\text{ such that }x_i^2\mid \mathbf{m}\text{ for all }i\right\}.$$

The formula is motivated by the relation with Stanley–Reisner complexes of polarizations of powers of edge ideals, but a proof that the existence of such a generator forces stabilization is not known. Formulas for stabilization indices of other monomial ideals also remain open.