The negativity conjecture for coefficients of the matching polynomial

Let $n$, $d$, $Y$, $M$, $f_r^{\pm}$, and $\lambda$ be as in the following definitions. Set

$$Y=\sum_M(-1)^M T_d(\{f_1^+,f_1^-,\ldots,f_{k/2}^+,f_{k/2}^-\},a),$$

where $M=\{(i_1<j_1),\ldots,(i_{k/2}<j_{k/2})\}$ ranges over all maximal matchings of $\{1,\ldots,n-1\}$ and $f_r^{\pm}=(e_{i_r}\pm e_{j_r})/\sqrt{2}$. For a partition $\lambda=(\lambda_1,\ldots,\lambda_{n-1})$ of $(d-n(n-1))/2$ into $n-1$ parts, define

$$\mu=a_1^{2\lambda_1+2(n-1)}a_2^{2\lambda_2+2(n-2)}\cdots a_{n-1}^{2\lambda_{n-1}+2}.$$

Matching-coefficient negativity conjecture. The coefficient of $\mu$ in $Y$ is negative. This coefficient-level assertion is used to establish the predicted sign in type $D_n$; it is supported by the matching expansion and symmetry arguments in the source, but is not resolved there.