Positivity conjecture for the Laurent polynomials \mathscr{L}_{k,n}^{d}$z$

For integers $k,d\ge1$ and $n\ge0$, let $\mathscr{L}_{k,n}^{d}(z)$ be the Laurent polynomials defined by the generating function in the source. Write

$$\mathscr{L}_{k,n}^{d}(z)=\sum_{-n-k\le m\le n+k}J_{k,n}^{d}(m)z^m.$$

Positivity conjecture. For all integers $k\ge1$, $d\ge2$, and $n\ge0$, the Laurent polynomial $\mathscr{L}_{k,n}^{d}(z)$ has positive coefficients. This conjecture is presented as a further refinement of the truncated Jacobi theta-series conjecture; its resolution is not specified in the supplied text.