Gambler's Ruin with Momentum (Correlated Steps)

A gambler on $\{0, 1, \ldots, 8\}$ has absorbing barriers at $0$ and $8$. Her step probabilities depend on the previous outcome: after a win ($+1$), she wins again with probability $2/3$ (momentum); after a loss ($-1$), she wins with probability $1/3$. The first step is a win with probability $1/2$. Starting from state $4$, what is the probability of ruin (reaching $0$)?