The reversal symmetry conjecture for expected coin-flip ending times

Let $S$ be a finite ending string of heads and tails, let $S'$ be its reversal, and let $E(S)$ denote the expected number of fair-coin flips needed to produce $S$. Thus, if $S=s_1s_2\cdots s_r$, then $S'=s_rs_{r-1}\cdots s_1$.

Reversal symmetry conjecture. If $S'$ is the reversal of an ending string $S$, then

$$E(S)=E(S').$$

The paper establishes this symmetry for ending strings with two, three, and four alternating runs through the cited preceding theorems. The conjecture extends that observed symmetry to arbitrary ending strings.