Chordal graph conjecture for square-free powers

Let $G$ be a chordal graph, let $I(G)$ be its edge ideal, let $I(G)^{[k]}$ denote its $k$-th square-free power, let $\nu(G)$ be its matching number, and let $\mathrm{aim}(G,k)$ be its $k$-admissible matching number. Chordal graph conjecture. For all $1\leq k\leq\nu(G)$, one has

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

The equality is known for block graphs and for $k=2$ when $G$ is Cohen–Macaulay and chordal; computations also support it for chordal graphs with at most eight vertices. The assertion remains open in general.