Oğuz–Ravichandran unimodality conjecture for restricted fence rank sequences

Let $F(\beta)$ be a fence with an even number of parts, and let $\overline{r}(\beta)$ denote the restricted rank sequence defined in the source. Oğuz–Ravichandran's conjecture. If $\beta=(\beta_1,\beta_2,\ldots,\beta_{2\ell})$, then $\overline{r}(\beta)$ is unimodal except when $\beta=(1,k,1,k)$ or $(k,1,k,1)$ for some $k\ge 1$. The conjecture concerns the exceptional behavior of restricted rank sequences for even-part fences; the supplied passage asks for an injective proof and gives no resolution status.