The scaling and Brownian approximation conjecture for gambler's ruin probabilities

For positive integers $A,B,C$, let $P_{A,B,C}(\sigma)$ denote the first hitting probability of the permutation $\sigma\in S_3$ in the gambler's ruin walk, and let $P_{A,B,C}^{\rm BM}(\sigma)$ denote the corresponding Brownian motion extinction probability. For $n\ge2$, write $nA,nB,nC$ for the scaled initial state. Scaling and Brownian approximation conjecture. (a) For each $A,B,C\ge1$, $\sigma\in S_3$, and $n\ge2$,

$$\left|P_{nA,nB,nC}(\sigma)-P_{A,B,C}(\sigma)\right|<0.0004.$$

(b) For each $A,B,C\ge1$, with $N:=A+B+C$, and each $\sigma\in S_3$,

$$\left|P_{nA,nB,nC}(\sigma)-P_{A,B,C}^{\rm BM}(\sigma)\right|=O\left(\frac{1}{(nN)^4}\right)\quad\text{as }n\to\infty.$$

These assertions are intended to explain the rapid convergence of the rescaled probabilities and provide a more precise version of the scaling conjecture. The stated bounds are conjectural, and the supplied context gives no resolution status.