In the paper by Greene, Nijenhuis and Wilf, an algorithm is proposed for generating uniformly random Young tableaux of shape $\lambda$. The algorithm is to uniformly randomly pick a starting cell, and then do a hook walk algorithm until it terminates at one of the edges of the tableau. Another way of looking at this is to fix a starting cell $(a,b)$, and then start a random hook walk, so that one gets a path

$$(a,b)=(a_1,b_1)\rightarrow(a_2,b_2)\rightarrow\cdots\rightarrow (a_k,b_k)=(\alpha,\beta)$$

where $(\alpha,\beta)$ is the terminal cell. This defines a probability of hitting terminal cell $(\alpha,\beta)$ given starting cell $(a,b)$.

A peculiar observation is made on page 108, that:

$$P(\ (\alpha,\beta) \ |\ (a, b)\ ) = P(\ (\alpha,\beta) \ | \ (\alpha, b)\ ) \cdot P(\ (\alpha,\beta)\ |\ (a,\beta) \ )$$

In other words the probability of reaching the terminal point is the product of probabilities of starting in the row and column of the terminal point, which amounts to perpetually staying within the respective hooks. The authors point out that they have no obvious direct explanation of this fact.

Has anyone come up with an explanation more recently?