Bangachev's minimum-degree conjecture for friends-and-strangers graphs

Let $X$ and $Y$ be connected graphs on $n$ vertices. For a graph $G$, write $\delta(G)$ for its minimum degree, and let $\operatorname{FS}(X,Y)$ denote the friends-and-strangers graph of $X$ and $Y$. Bangachev's conjecture. If

$$2\min(\delta(X),\delta(Y))+3\max(\delta(X),\delta(Y))\geq 3n,$$

then $\operatorname{FS}(X,Y)$ is connected. This conjecture concerns minimum-degree conditions guaranteeing connectedness of friends-and-strangers graphs; the supplied text attributes it to Bangachev but gives no evidence of resolution.