Characterization of graphs for which the blow-up optimization is unattained

Let $H$ be a graph with no isolated vertices, let $k$ be a positive integer, and let $\beta(H,k)$ be the optimization parameter defined in the paper. Attainment conjecture for $\beta(H,k)$. The quantity $\beta(H,k)$ is not achieved if and only if $k=1$ and either $H=K_{1,m}$ or $H=mK_2$ for some $m\geq 2$, where $mK_2$ is the matching on $m$ edges.

The paper notes that the two listed families are known never to attain the optimum, whereas $\beta(H,k)$ is attained whenever $k\delta(H)\geq 2$. The conjecture proposes that these are the only exceptions.