The connected 3-colouring conjecture for general uniform hypergraphs

Let $K_n^{(r)}$ be the complete $r$-uniform hypergraph on $n$ vertices. Call an $r$-uniform hypergraph connected when there is a strong path between every pair of $(r-1)$-sets, where consecutive edges in a strong path intersect in exactly $r-1$ vertices; call a colouring of $K_n^{(r)}$ connected when each colour class spans a connected subhypergraph. An $(r+1)$-set uses all three colours if its $r$-edges include edges of each of the three colours. Connected 3-colouring conjecture. For all sufficiently large $n$, if the edges of $K_n^{(r)}$ are connectedly 3-coloured, then there must exist an $(r+1)$-set that uses all three colours. The paper identifies the three-colour case as the interesting generalisation to $r$-uniform hypergraphs, but provides no proof of this assertion.