The Sturm-sequence conjecture for $(q,t)$-Hoggatt sums

For integers $q>0$ and $m=2,3$, let $H_n^{[m]}(q,t)$ denote the $(q,t)$-Hoggatt sums. The Sturm-sequence conjecture. The polynomial sequence

$$\{H_n^{[m]}(q,t)\}_{n\geq 1}$$

forms a Sturm sequence. The authors verified this computationally for $1\leq q\leq 10$ and $1\leq n\leq 15$; the case $q=1$ is established by the paper's theorem, while the general claim remains open.