Higher-uniformity extremal construction conjecture for tight cycles

Fix a uniformity $r\geq 2$. Let $C_L^{(r)}$ be the $r$-uniform tight cycle of length $L$, and call an $r$-graph $C_L^{(r)}$-hom-free when it contains no homomorphic copy of this cycle. A complete oddly bipartite $r$-graph is the construction in which an edge is present precisely when it meets the relevant bipartition in an odd number of vertices.

Higher-uniformity extremal construction conjecture. For all sufficiently long $L$ relatively prime to $r$, the extremal $C_L^{(r)}$-hom-free constructions on sufficiently large numbers of vertices are the complete oddly bipartite $r$-graph when $r$ is even, and the iterated blowup described in the source when $r$ is odd.

This conjecture extends the proposed extremal constructions from uniformities $5$ and $6$ to arbitrary uniformity. It remains open, and the source notes that higher-uniformity analogues of the available structural arguments would require new ideas.