Connectivity conjecture for Hamilton cycles in hypergraph line graphs

For an integer $r\geq 2$, a rank $r$ hypergraph is a hypergraph whose edges have size at most $r$. Its line graph is the graph whose vertices represent hypergraph edges, with adjacency when the corresponding hyperedges intersect. A graph is $\phi(r)$-connected if deleting fewer than $\phi(r)$ vertices leaves it connected.

Hypergraph line-graph connectivity conjecture. For any $r\geq 2$, there is an integer $\phi(r)$ such that every $\phi(r)$-connected line graph of a rank $r$ hypergraph is Hamiltonian.

The conjecture was recently proposed in the cited work and is open in the generality stated. The paper proves that every $52$-connected line graph of a rank $3$ hypergraph is Hamiltonian, which is partial progress toward the conjecture.