Apex conjecture for connected bipartite graphs satisfying Sidorenko's conjecture

Let $H$ be a connected bipartite graph satisfying Sidorenko's conjecture. For a positive integer $a$, let $H^{+a}$ be the graph obtained from $H$ by adding $a$ vertices, each adjacent to every vertex of $H$, with no edges among the added vertices. Call a graph common when the random 2-edge-colouring asymptotically minimises its monochromatic-copy count.

Apex conjecture. For every positive integer $a$, the graph $H^{+a}$ is common. In particular, every complete tripartite graph $K_{r,s,t}$ is common.

The paper proves commonality for adding arbitrarily many apex vertices when $H$ is a connected bipartite graph on at most five vertices, but the stated general assertion remains unresolved in the supplied text.