Hyperness conjecture for generalized dissections

Let ${\mathcal{D}}$ be a generalized dissection, let ${\mathcal{T}}({\mathcal{D}})$ be its associated constrained triangulation, and let $V({\mathcal{D}})$ and $Y({\mathcal{D}})$ denote the associated area variety and projective area space. Hyperness conjecture. The triangulation ${\mathcal{T}}({\mathcal{D}})$ is hyper; equivalently, $V({\mathcal{D}})$ is a hypersurface in $Y({\mathcal{D}})$.

The claim asserts that generalized dissections always produce area varieties of codimension one. The supplied text gives no evidence that this conjecture has been proved or disproved.