Deletion–triangle subdivision conjecture for Hilbert bases of cuts

Let $G$ be a graph in the class $\mathscr{H}$ of graphs whose cut semigroup has the Hilbert basis property, and let $f\in E(G)$ be an edge. Write $G\setminus f$ for deletion of $f$, and $G+_f C_3$ for the graph obtained by replacing $f$ with a triangle. Deletion–triangle subdivision conjecture. Then

$$G\setminus f\in\mathscr{H}\quad\text{if and only if}\quad G+_f C_3\in\mathscr{H}.$$

The preceding theorem establishes the analogous equivalence with $C_n$ for every $n\geq 4$; the case of subdivision by a triangle is left as a conjecture because one direction of that proof breaks down when the edge is subdivided only once.