Conjecture on non-Hamiltonian graphs with an arbitrarily high fraction of Hamiltonian vertex pairs

Let $G$ be a graph, and let $r_G$ denote the fraction of pairs of vertices of $G$ that are connected by a Hamiltonian path. Equivalently, if $G$ has order $n$, define

$$r_G = \max\left\{\frac{k}{\binom{n}{2}}\;\middle|\;G\text{ is }k\text{-pair strung}\right\},$$

where $G$ is $k$-pair strung when $k$ pairs of vertices can be connected by a Hamiltonian path. The conjecture. For every $\epsilon>0$, there exists a graph $G$ such that

$$r_G>1-\epsilon.$$

The conjecture predicts graphs in which an arbitrarily large fraction of vertex pairs are connected by Hamiltonian paths, despite the broader discussion concerning non-Hamiltonian graphs. The paper describes constructions whose corresponding fractions approach $4/5$ and $6/7$; finding graphs with arbitrarily large H-path connected edge sets would establish the stated bound.