Sign conjecture for the abstract matrix-tree Laplacian

Let $P_m \stackrel{\text{def}}{=} ([1,2],[1,3],\dots,[1,m])\in\Gamma_{n,m}$. For any $k$, define

$$X(\{G\in\Gamma_{n,k}\mid P_m*G\in\mathop{\rm \mathfrak S}\nolimits_{n,k+m}^\varnothing\})$$

and write its Laplacian as an element of the graph space:

$$\Delta\bigl(X(\{G\in\Gamma_{n,k}\mid P_m*G\in\mathop{\rm \mathfrak S}\nolimits_{n,k+m}^\varnothing\})\bigr)=\sum_{H\in\Gamma_{n,k}}x_HH.$$

Sign conjecture. All coefficients $x_H$ are integers of the same sign.

This conjecture is presented as a generalization of the paper's codimension-one theorem. The source provides explicit calculations motivating the claim, but the supplied text gives no resolution, so its status remains open.