Conjecture on unique Hamiltonian cycles in $P_4+sP_1$-free graphs

Let $s\geq 1$ be an integer. Let $P_k$ denote the path on $k$ vertices, and let $sP_1$ denote the disjoint union of $s$ isolated vertices. A graph is $P_4+sP_1$-free if it has no induced subgraph isomorphic to $P_4+sP_1$. Let $G$ be a Hamiltonian graph on $n$ vertices, meaning that $G$ contains a spanning cycle. It is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle.

Conjecture. There exists a constant $n_s$ depending on $s$ such that $G$ is not uniquely Hamiltonian whenever $n\geq n_s$.

This conjecture predicts that, for each fixed $s$, sufficiently large Hamiltonian $P_4+sP_1$-free graphs must contain at least two Hamiltonian cycles. It is posed as a direction for future work, and the supplied text gives no resolution status.