NP-hardness of Circuit Distance for polygons

Let $P = \{\mathbf{x}\in \mathbb{R}^n\colon A\mathbf{x}\leq \mathbf{b}\}$ be a polytope defined by a matrix $A\in \mathbb{Q}^{m\times n}$ and a vector $\mathbf{b}\in \mathbb{Q}^m$, let $\mathbf{s}$ and $\mathbf{t}$ be vertices of $P$, and let $k\in \mathbb{Z}_{\geq 0}$. A circuit walk is a walk from $\mathbf{s}$ to $\mathbf{t}$ whose steps follow circuit directions of $P$. The conjectured decision problem asks whether there is a circuit walk from $\mathbf{s}$ to $\mathbf{t}$ of length at most $k$.

Circuit Distance conjecture. The Circuit Distance problem is NP-hard for polygons.

The paper notes that the proof techniques for monotone circuit walks seem likely to extend to the undirected setting but require technical innovation; the claimed NP-hardness for polygons therefore remains open in the supplied source.