Eisenbud–Huneke–Ulrich conjecture on powers of ideals with linear syzygies

Let $S=\mathbb{C}[x_0,x_1,\dots,x_n]$, let $\mathfrak{m}$ be the maximal ideal generated by the variables, and let $I$ be a homogeneous ideal generated in degree $d$. The resolution of $I$ is linear for $p$ steps if its first linear strand is exact up to cohomological degree $-p+1$, equivalently, for every $0\leq i\leq p$, one has

$$\beta_{i,j}(I)=\dim_{\mathbb{C}}\left(\operatorname{Tor}_i^S(I,\mathbb{C})_{i+j}\right)\neq 0\quad\Longleftrightarrow\quad j=d.$$

Eisenbud–Huneke–Ulrich conjecture. If $I$ is $\mathfrak{m}$-primary and the resolution of $I$ is linear for $p$ steps, where $p\geq 1$, then

$$I^t=\mathfrak{m}^{td}\quad\text{for all }t\geq\frac{n}{p}.$$

The conjecture predicts that sufficiently many initial linear syzygy steps force every sufficiently high power of an $\mathfrak{m}$-primary ideal to equal the corresponding power of the maximal ideal. The paper proves a slightly weaker version for a more general class of ideals.