The large bond-dimension conjecture for maximal flattening rank on 2 by N grids

Let $G_{2 \times N}$ be the tensor network graph with $2N$ physical edges, and let $d,k \in \mathbb{N}$. Write $\operatorname{TNS}(G_{2 \times N},d,k)$ for the associated tensor network states, with physical dimension $d$ and bond dimension $k$. A tensor has maximal flattening rank when every flattening attains the largest rank allowed by the quantum max-flow/min-cut inequality.

Large bond-dimension conjecture. For $d=2$ and $k$ sufficiently large, there exists a tensor $T \in \operatorname{TNS}(G_{2 \times N},d,k)$ such that all flattenings of $T$ have maximal rank, as constrained by the quantum max-flow/min-cut inequality.

This conjecture asks when tensor network states contain tensors simultaneously attaining the quantum max-flow/min-cut bound for every flattening. Such tensors would show that, in this setting, the tensor network state variety reaches the corresponding quantum max-flow/min-cut constraints; the source presents the claim as motivated by empirical findings, without providing a resolution.