The alternating two-player urn distribution conjecture

Consider two walkers on the three-node segment, both starting at the center, with both edge weights initially equal to $1$. The walkers move alternately, walker $1$ first, and each center departure chooses an edge with probability proportional to its current edge weight. Let $w(n,0)$ and $w(n,1)$ be the left and right edge weights after time $n$, and define

$$Y:=\lim_{n\to\infty}\frac{w(n,0)}{w(n,0)+w(n,1)}.$$

Alternating two-player urn distribution conjecture. The random variable $Y\in[0,1]$ has a density with respect to Lebesgue measure on $[0,1]$, is not Beta-distributed, and has a distribution different from that of the limit in the corresponding non-alternating two-player model.

The alternating model has a martingale only at suitable stopping times, but its edge-weight fraction is known to converge almost surely. The density and distributional comparisons are suggested by simulations and remain unproved.