Refinement-order compatibility under T-duality

Consider regular positroidal subdivisions $\{\Gamma_{\pi}\}$ and $\{\Gamma_{\pi'}\}$ of $\Delta_{k+1,n}$ and corresponding subdivisions $\{Z_{\hat{\pi}}\}$ and $\{Z_{\hat{\pi'}}\}$ of $\mathcal{A}_{n,k,2}$. Write $\{\Gamma_{\pi}\}\preceq\{\Gamma_{\pi'}\}$ when the first subdivision refines the second, and similarly for the $Z$-subdivisions. Refinement-order compatibility conjecture. One has $\{\Gamma_{\pi}\}\preceq\{\Gamma_{\pi'}\}$ if and only if $\{Z_{\hat{\pi}}\}\preceq\{Z_{\hat{\pi'}}\}$. This asserts that T-duality preserves the refinement poset of regular subdivisions.