Random-walk super-coupon collection conjecture

Let the universe consist of $n$ coupons, and let $H_t$ be an $r$-subset drawn at round $t$. Starting from $H_t$, form $H_{t+1}$ by choosing a uniformly random coupon in $H_t$ and replacing it with a uniformly chosen coupon outside $H_t$. Let $T^{(r,s)}_{\mathrm{RW}}$ be the time needed to collect all $s$-subsets of coupons, where every $s$-subset of each drawn $r$-subset is collected. Random-walk super-coupon collection conjecture. As $n\to\infty$, the expected collection time satisfies

$$\mathbb{E}T^{(r,s)}_{\mathrm{RW}}=\frac{\frac{r}{s}\binom{n}{s}\log\binom{n}{s}}{\binom{r}{s}}(1+o(1)).$$

The independent-draw model has the corresponding expectation without the factor $r/s$, while the random-walk model is connected to cover times of random walks on finite regular graphs. The conjecture is motivated by simulations; its validity for this temporally dependent model remains open.