Shellability conjecture for the DKK-triangulation of the flow polytope

Let $obreak\root$ be the DKK-triangulation $obreak\root_n^{\textup{LRL}}$ of the flow polytope associated with the zigzag graph. The modified lexicographic ordering on its simplices is obtained by first ordering their coordinate vectors by the sum of their coordinates and then ordering vectors with equal sums lexicographically; let the reverse ordering be the opposite ordering.

Shellability conjecture. The DKK-triangulation $obreak\root_n^{\textup{LRL}}$ is shellable because both the modified lexicographic ordering and its reverse are shellings.

If true, the two shellings provide combinatorial interpretations of the coefficients of the $h$-polynomial, and hence of the $h^*$-polynomial because the triangulation is unimodular. The source gives no resolution of this conjecture.