Unimodality conjecture for minimal obstruction counts of uniform clutters

For a $d$-uniform clutter $\mathcal{C}$, let $I(\mathcal{C})$ be its edge ideal, and let $\Omega_{d,k}$ consist of the clutters satisfying

$$\operatorname{index}(I(\mathcal{C})),\ldots,\operatorname{index}(I(\mathcal{C})^{k-1})>1,$$ $$\operatorname{index}(I(\mathcal{C})^k)=1,$$

and having no proper induced sub-clutter satisfying these two conditions. Let $\Omega_{d,k}(n)$ be the number of isomorphism classes in $\Omega_{d,k}$ with $n$ vertices. Unimodality conjecture. The sequence $\{\Omega_{d,k}(n)\}_n$ is unimodal. The conjecture proposes a unimodal distribution of minimal uniform-clutter obstructions according to their number of vertices; the examples listed in the source exhibit the claimed pattern for several small values of $d$ and $k$, while no general proof or resolution is given.