The tightly connected monochromatic subgraph conjecture for complete uniform hypergraphs

Let $K_n^r$ denote the complete $r$-uniform hypergraph on $n$ vertices. A tightly connected subgraph is one in which every two vertices can be joined by a sequence of edges in which consecutive edges intersect in exactly $r-1$ vertices.

Tightly connected hypergraph conjecture. For every $r\geq 3$, every $r$-coloring of $K_n^r$ contains a monochromatic tightly connected subgraph covering all vertices of $K_n^r$.

The paper proves the case $r=3$; the general statement remains open.