Infinite q-log-convexity of the polynomial sequences in Example basic-qSM

Let $\{f_i(q)\}_{i\geq 0}$ be any of the polynomial sequences defined in Example $\mathrm{basic\text{-}qSM}$. A sequence is infinitely $q$-log-convex if every iterate under

$$\mathcal{L}\{f_i(q)\}_{i\geq 0}=\{f_{i-1}(q)f_{i+1}(q)-f_i(q)^2\}_{i\geq 1}$$

has coefficientwise nonnegative polynomials. The infinite $q$-log-convexity conjecture. Every sequence of polynomials in Example $\mathrm{basic\text{-}qSM}$ is infinitely $q$-log-convex. The paper notes that these sequences are already known to be $3$-$q$-log-convex and, for fixed nonnegative real $q$, infinitely log-convex; the infinite $q$-analogue remains proposed for further research.