Unimodality conjecture for rank polynomials of circular fence posets

Let $\bar{a}$ be a tuple of positive integers, and let $\overline{F}(\bar{a})$ be the corresponding circular fence poset with rank polynomial $\overline{R}(\bar{a};q)$. A polynomial is unimodal if its coefficients weakly increase up to a point and then weakly decrease. Circular-fence rank-polynomial conjecture. The rank polynomial $\overline{R}(\bar{a};q)$ of a circular fence poset $\overline{F}(\bar{a})$ is unimodal except when

$$\bar{a}=(a,1,a,1)\quad\text{or}\quad(1,a,1,a)$$

for some positive integer $a$. This conjecture concerns the apparent general unimodality of rank polynomials, with the displayed families as the proposed exceptions; the source reports that extensive computer calculations suggested it, but gives no resolution.