Hereditary characterization conjecture for ${\cal G}_k$

Let $k$ be a positive integer, let ${\cal G}_k$ be the graph class defined in the paper, and for a graph $H$ let $\nu_k(H)$ denote the maximum number of pairwise vertex-disjoint $k$-paths and let $\tau_k(H)$ denote the minimum size of a vertex set meeting every $k$-path. A subgraph need not be induced.

Hereditary characterization conjecture. For every positive integer $k$, the set ${\cal G}_k$ equals the set of all graphs $G$ such that

$$\nu_k(H)=\tau_k(H)$$

for every subgraph $H$ of $G$.

The claim generalizes observations proved in the paper for $k\in\{1,2,3,4\}$. It gives a proposed characterization of ${\cal G}_k$ through the equality of maximum $k$-matchings and minimum $k$-vertex covers in every subgraph, but remains open for general positive integer $k$.