Linear average-case performance conjecture for MemberPN

Let $w$ be a binary word, and let ${\sc MemberPN}(w)$ be the proposed membership tester that applies the two linear-time rejection tests followed by a quadratic-time prefix-normality test. MemberPN average-case conjecture. The membership tester ${\sc MemberPN}(w)$ for prefix normal words functions in average-case $O(n)$ time. The conjecture is motivated by decreasing empirical ratios in the reported experiments, but remains unproved in the source.