Generalized Hamilton-connectedness conjecture for forbidden-star split graphs

Let $r\geq 3$ be an integer. A graph is split if its vertex set can be partitioned into a clique and an independent set. It is ${K_{1,r+1},K_{1,r+1}+e}$-free if it contains no induced subgraph isomorphic to either forbidden graph, where $K_{1,r+1}+e$ is obtained from $K_{1,r+1}$ by adding an edge joining two nonadjacent vertices. A graph is Hamilton-connected if every two distinct vertices are joined by a Hamiltonian path.

Generalized Hamilton-connectedness conjecture. Every $r$-connected ${K_{1,r+1},K_{1,r+1}+e}$-free split graph of order at least $2r+7$ is Hamilton-connected.

The conjecture is motivated by the paper's theorem for $r=3$, namely that every $3$-connected ${K_{1,4},K_{1,4}+e}$-free split graph of order at least $13$ is Hamilton-connected. The general case is posed for further study and is unresolved in the supplied text.