The FDH optimality conjecture for m-shelf shuffles

Let $H_{2m}$ be the $2m$th harmonic number, and let $F^{\mathbf{G}}(n,m)$ denote the expected number of correct guesses made by the strategy $\mathbf{G}$ for a deck of $n$ cards subjected to an $m$-shelf shuffle. Assume that $n/m$ is not too small. FDH optimality conjecture. The strategy $\mathbf{G}$ discussed by FDH should be optimal in a high-probability case, and its expected number of correct guesses satisfies

$$F^{\mathbf{G}}(n,m)\approx\frac{n}{2m}H_{2m}=\frac{n}{2m}\sum_{k=1}^{2m}\frac{1}{k}.$$

The source motivates this as a generalization of the one-shelf result and notes its applicability when the ratio of cards to shelves is not too small, but the meaning of “high-probability case” and the approximation are not made precise in the supplied statement.