Connected transversal threshold conjecture for complete multipartite graphs

Let $r\geq 4$, let $K_r$ be the complete graph on $r$ parts, and let $\mathcal{T}_r$ denote the family of all trees on $r$ vertices. For an $r$-partite graph, write $\pi_{K_r}(\mathcal{T}_r)$ for the smallest $r$-partite density forcing a connected transversal tree.

Connected transversal conjecture. The lower bound in Theorem~ is tight. In particular, for any $r\geq 4$, every $r$-partite graph with $r$-partite density at least $\frac{3-\sqrt{5}}{2}$ contains a transversal tree on $r$ vertices.

The conjecture would determine the connected-transversal density threshold for complete multipartite graphs, improving the presently established bounds.