Gatzweiler–Krattenthaler positivity conjecture for quotients of Gaussian binomials

Let $n,k,l\in\operatorname{\mathbb{Z}}_{\geq 0}$, and write $[r]_q!=\prod_{i=1}^r[i]_q$. Consider

$$\frac{\binom{n}{k}_q}{\binom{n}{l}_q}=\frac{[l]_q![n-l]_q!}{[k]_q![n-k]_q!}=\frac{\prod_{i=1}^l(1-q^i)\prod_{i=1}^{n-l}(1-q^i)}{\prod_{i=1}^k(1-q^i)\prod_{i=1}^{n-k}(1-q^i)}.$$

Gatzweiler–Krattenthaler conjecture. If this quotient is a polynomial, then it has non-negative integer coefficients. The conjecture is a positivity assertion for polynomial quotients of Gaussian binomial coefficients. The source reports that it is proved in several special cases, including $l=k,k-1,k-2,k-3$ and $l=0,1,2$, but leaves the general statement unresolved.