The powers-of-two conjecture for expected coin-flip ending times

Let $S$ be a finite string of heads and tails, let $s$ be its total number of heads and tails, and let $E(S)$ denote the expected number of fair-coin flips needed to produce $S$. Assume

$$S\notin\{\mathrm{H}^k,\mathrm{T}^k\}.$$

Powers-of-two conjecture. The expected value $E(S)$ equals $2^s$ possibly plus some lower positive powers of $2$; asymptotically, $E(S)$ is about $2^s$.

The paper proves this form of behavior for strings with at most four maximal runs or for alternating strings, and the conjecture proposes it for arbitrary ending strings. An intuitive explanation for the resulting sums of powers of $2$ remains to be found.