Unimodality conjecture for Betti numbers of venustus real toric manifolds

Let $\mathcal{B}$ be a venustus building set, and let $X^\mathbb{R}_{\mathcal{B}}$ be its associated real toric manifold. Write $\beta_k(X^\mathbb{R}_{\mathcal{B}})$ for its rational Betti numbers. Unimodality conjecture. The sequence

$$\{\beta_k(X^\mathbb{R}_{\mathcal{B}})\}_{k \geq 0}$$

is unimodal. The authors report that unimodality holds for the real permutohedral varieties and that they have found no graphical building set whose Betti-number sequence is not unimodal; the conjecture remains open for venustus building sets in general.