Hamilton path conjecture for connected graphs with exactly two spanning paths after vertex deletion

Let $G$ be a connected graph on $n\geq 2$ vertices. Here, $\psi_k(G)$ denotes the number of paths in $G$ with $k$ vertices.

Hamilton path conjecture. The following implication holds

$$\psi_{n-1}(G)=2 \Rightarrow \psi_n(G)=1.$$

The conjecture asserts that a connected graph with exactly two paths on $n-1$ vertices must contain a Hamilton path. It was proposed from computational data for graphs with up to nine vertices, and the source states that, to the authors' knowledge, it had not previously been studied.