Strong Schur log-concavity conjecture for diagonals of the quantum Pascal triangle

For $n\geqslant k\geqslant0$, define the quantum binomial by

$$\genfrac{[}{]}{0pt}{}{n}{k}=\frac{[n][n-1]\cdots[n-k+1]}{[k][k-1]\cdots[1]},$$

where $[n]=(q^{-n}-q^n)/(q^{-1}-q)$ and $[0]=0$. A sequence of Laurent polynomials is strongly Schur log-concave if each difference $a_{n+i}(q)a_n(q)-a_{n+i+1}(q)a_{n-1}(q)$ is centred, unimodal, and has non-negative coefficients. Quantum Pascal diagonal conjecture. For all choices of $n\geqslant k\geqslant0$, $\alpha\geqslant-1$, and $\beta\geqslant0$, the diagonal

$$\genfrac{[}{]}{0pt}{}{n}{k},\quad \genfrac{[}{]}{0pt}{}{n-\alpha}{k+\beta},\quad \genfrac{[}{]}{0pt}{}{n-2\alpha}{k+2\beta},\quad \genfrac{[}{]}{0pt}{}{n-3\alpha}{k+3\beta},\quad \genfrac{[}{]}{0pt}{}{n-4\alpha}{k+4\beta},\quad\ldots$$

of slope $\alpha/\beta$ is strongly Schur log-concave. This is a data-driven conjecture about quantum analogues of Pascal-triangle diagonals; no resolution is supplied in the stated material.