The connected 3-colouring conjecture for tricoloured 4-sets

Let $K_n^{(3)}$ be the complete 3-uniform hypergraph on $n$ vertices, and call a colouring connected when each colour class spans a connected subhypergraph. A 4-set is tricoloured if its four 3-edges use all three colours. Connected 3-colouring conjecture. For all sufficiently large $n$, every connected 3-colouring of $K_n^{(3)}$ must contain a tricoloured 4-set. The paper constructs covering colourings with no tricoloured 4-set when the number of colours is increased inductively, while this connected 3-colour case is posed as an extension of Gallai's theorem and remains open.