BCFW triangulation conjecture for amplituhedra in four dimensions

For $m=4$, let $\mathcal{T}_{k,\ell}=\{f_1,\dots,f_N\}$ be the BCFW collection of positroid cells, with $\mathcal{T}_{\ell,k}=\{f_1^{-1},\dots,f_N^{-1}\}$ up to cyclic shift. BCFW triangulation conjecture. For every $k$ and $\ell$, $\mathcal{T}_{k,\ell}$ is an $(n,k,m)$-triangulation of degree $1$. The BCFW collection is widely used to compute the amplituhedron form; the source gives no general proof.