Bowler–Brown–Fenner common-card bound for trees and connected non-trees

Let $G$ and $H$ be graphs on $n$ vertices, where one is a tree and the other is a connected non-tree. A common card is an unlabeled graph occurring as a card in the decks of both $G$ and $H$. Bowler–Brown–Fenner conjecture. For $n\geq 44$, the only pair of graphs on $n$ vertices having at least $\lfloor \frac{2}{5}(n+1)\rfloor$ common cards is the pair found in Theorem 3.6 of Bowler, Brown, and Fenner. This conjecture concerns the extremal overlap between the decks of trees and connected non-trees; the source does not state whether it has been resolved.