The regularity formula for squarefree powers of edge ideals of forests

Let $G$ be a forest. For a positive integer $k$, let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$, let $\operatorname{reg}$ denote Castelnuovo–Mumford regularity, let $\operatorname{aim}(G,k)$ denote the maximum cardinality of an induced $k$-matching of $G$, and let $\operatorname{mat}(G)$ denote the matching number of $G$.

Forest regularity conjecture. For every $1\leq k\leq \operatorname{mat}(G)$,

$$\operatorname{reg}(I(G)^{[k]})= \operatorname{aim}(G,k)+k.$$

The conjecture proposes that the upper bound established for squarefree powers of edge ideals of forests is always sharp. It is motivated by the preceding results and Macaulay2 computations, but the source provides no resolution.