The bounded geodecity conjecture for subdivisions of graphs

Let $G$ be a graph. A subdivision of $G$ is a graph obtained by replacing edges of $G$ with paths. For each positive integer $m$, let \H_m denote the class of graphs whose geodecity is at most $m$.

Bounded geodecity conjecture. For every graph $G$ there exists an integer $m$ such that every subdivision of $G$ lies in \H_m.

This statement is presented as a consequence of a possible qualitative converse to the result that graphs in \H_k exclude $K_{2,k+2}$ as a minor. The paper does not establish it, so its status remains open.