Positivity conjecture for the top-defect matrix coefficient

Let $k\geq 1$, $0\leq h\leq k$, and let $g$ lie in the redundancy range $3g\geq 3k+h+1$. Let $W_{k,h}(g,3g-k-h-2)$ denote the matrix coefficient defined in the source's construction of the pairing.

Positivity conjecture. In the redundancy range,

$$W_{k,h}(g,3g-k-h-2)>0.$$

The conjecture was prompted by extensive numerical computations. The source separately records a proof for $g\geq k+h+2$, while the assertion in the full redundancy range is not otherwise resolved in the supplied text.