Expected number of draws to obtain a soft streak

Let consecutive letters be drawn uniformly from the alphabet $\{1,2,\ldots,n\}$, and stop when a soft streak of length $k$—that is, a non-decreasing word of length $k$—is obtained. Let $\psi_{k,r}$ and $\omega_k$ be the quantities used in the preceding generating-function identities. Then the expected number of letters drawn is

$$E_{\mathrm{soft}}(n,k)=\frac{(1-n^{-k})^n}{\displaystyle\sum_{r=0}^{(k-1)n}\psi_{k,r}\operatorname{B}_s(n,k,r)n^{-r}}$$

and equivalently

$$E_{\mathrm{soft}}(n,k)=\frac{k}{\displaystyle\sum_{s=1}^{k-1}(1-\omega_k^{-s})\left(1-\frac{\omega_k^s}{n}\right)^{-n}}.$$

Expected-number conjecture. The expected number of letters drawn is given by these two equivalent expressions.

This is presented as a corollary of the generating-function method and extends the preceding theorem for strictly increasing streaks to soft streaks. The supplied excerpt does not provide evidence that the conjecture has been resolved.