The component-star conjecture for disconnected graphs

Let $G$ be a disconnected graph whose components are all isomorphic to a graph $H$, and let $ern(G)$ be the minimum number of edge-cards needed to determine $G$ up to isomorphism. Component-star conjecture. If $ern(G)>3$, then $H$ is isomorphic to the star $K_{1,r}$, where $r$ is the number of edges of $H$.

The conjecture is attributed in the surrounding discussion to Asciak and Lauri and is motivated by computational evidence and bounds for disconnected graphs. The supplied text does not report a resolution.