The unimodality conjecture for q-rational numerator and denominator polynomials

Let $\left[\frac{r}{s}\right]_{q}=\frac{\mathcal{R}(q)}{\mathcal{S}(q)}$ be a $q$-rational, where $\mathcal{R}(q)$ and $\mathcal{S}(q)$ are its numerator and denominator polynomials. Unimodality conjecture. For every $\left[\frac{r}{s}\right]_{q}=\frac{\mathcal{R}(q)}{\mathcal{S}(q)}$, the coefficients of the polynomials $\mathcal{R}(q)$ and $\mathcal{S}(q)$ form unimodal sequences. Long computer experimentation and evidence in the simplest cases motivate this conjecture; the supplied text gives no resolution, so its general validity remains open.