The unimodality-or-bimodality conjecture for restricted fractional-partition generating functions

Let $r(x)$ be the generating function whose coefficients $f_{O_j}(k)$ count the relevant partitions into fractions with fixed denominator $j$, with $k$ ranging over the integer values considered in the paper. A coefficient sequence is unimodal if it increases to a maximum and then decreases, and bimodal if it has two modes. Unimodality-or-bimodality conjecture. The generating function $r(x)$ is always either unimodal or bimodal.