Bistellar flip conjecture for amplituhedron triangulations

Let $g\in\tilde{S_n}(-k,n-k)$ be nearly $(n,k,m)$-admissible, meaning that $\operatorname{inv}(g)=k\ell-1$, $\operatorname{dim}(Z(\Pi^{>0}_{k+g}))=km$ for every $Z\in\operatorname{Gr}_{>0}(k+m,n)$, and each $h\lessdot g$ has either degree $1$ for every such $Z$ or degree $\infty$ for every such $Z$. Bistellar flip conjecture. There exist two disjoint sets $\mathcal{T}_g,\mathcal{T}'_g\subset\tilde{S_n}(-k,\ell)$ of the same size such that, for every $Z\in\operatorname{Gr}_{>0}(k+m,n)$, they are the only two triangulations of $Z(\Pi^{>0}_{k+g})$, and their union consists of all $h\lessdot g$ with degree $1$. This is the amplituhedron analogue of bistellar flips of polytopal triangulations; the source gives no resolution.