The one-third approval-ratio conjecture for double-interval societies

A double-interval society is a society in which each voter's approval set is the union of two intervals, and a society is pairwise-intersecting when every two voters' approval sets intersect. For such a society $S$, let $a(S)$ denote its approval ratio and let $n$ denote the number of voters.

One-third approval-ratio conjecture. For all pairwise-intersecting double-interval societies $S$, the approval ratio

$$\frac{a(S)}{n} \geq \frac13.$$

This conjecture is the authors' initial conjecture, for which they report additional supporting evidence. Its resolution is not established in the supplied text.