The secretary problem bound for posets with a fixed number of maximal elements

Let $(P,\prec)$ be a poset with $k$ maximal elements. Define

$$p_k=\begin{cases}\frac{1}{e},&k=1,\\[0.3ex]\sqrt[k-1]{\frac{1}{k}},&k>1.\end{cases}$$

Secretary problem conjecture. There is an algorithm for the secretary problem on $(P,\prec)$ that is successful with probability at least $p_k$.

The bound is proved in the source for posets of width $k$ and is expected to extend to all posets with $k$ maximal elements; it is not known from the supplied text whether the same algorithm works universally or another algorithm depending only on $k$ is required.