The lower-closure characterization of stably regular triangulations

Let $P$ be a planar point set, let ${\cal T}_{\mathsf{part}}(P)$ denote its partial triangulations, and let $\mathsf{S}_{\mathsf{triv}}$ be the trivial subdivision. A partial triangulation $T$ is stably regular if it is regular for every realization of the order type of $P$.

Stably regular triangulation conjecture. A triangulation $T \in {\cal T}_{\mathsf{part}}(P)$ is stably regular if and only if

$$T \prec^*_1 \mathsf{S}_{\mathsf{triv}}.$$

The claim proposes an order-type-based characterization of stable regularity, extending the paper's lower-closure criterion. It is presented as a belief about the strongest possible condition based only on the order type, and no resolution is given here.