Schlosser–Zeng conjecture on unimodality of truncated theta coefficients

For $k\in\mathbb N_0$, define the coefficients $J_k^{\textsc{t}}(m,n)$ by

$$\sum_{n\ge 0}\sum_{m\in\mathbb Z}J_k^{\textsc{t}}(m,n)q^nz^m =\frac{(-1)^k}{(z,q/z,q;q)_\infty}\sum_{0\le \ell\le k}(-1)^\ell q^{\ell(\ell+1)/2}z^{-\ell}(1-z^{2\ell+1}).$$

Schlosser–Zeng's conjecture. For every $k\in\mathbb N_0$ and $n\in\mathbb N$, the sequence

$$\bigl(J_k^{\textsc{t}}(m,n)\bigr)_{-n\le m\le n}$$

is unimodal.

The conjecture concerns the coefficient sequences arising from a truncated Jacobi triple product. The source gives no evidence of a resolution, so it remains open.