Let $S(n)$ be the discrete sphere of radius $n$ (i.e., the internal boundary of the Euclidean discrete ball $B(n)$) centered in the origin, and consider a simple random walk starting at some $x\in\mathbb{Z}^d$ with $2n-1<\|x\|\leq 2n$ (the dimension $d$ is at least 2). Let $\nu_{n,x}$ be the conditional entrance measure to $S(n)$, given that the walk hits $S(n)$ before $S(4n)$. Also, let $hm_n$ be the harmonic measure (from infinity) on $S(n)$. Can one prove that the R.-N. derivative of $\nu_{n,x}$ with respect to $hm_n$ satisfies the Lipschitz condition? That is, there is a constant $C$ (maybe depending on the dimension), such that $$\Big| \frac{\nu_{n,x}(y)}{hm_n(y)} - \frac{\nu_{n,x}(z)}{hm_n(z)} \Big| \leq C \frac{\|y-z\|}{n}, $$ for all $y,z\in S(n)$.
This is obviously true for the Brownian motion, but the discrete case is more tricky, since the harmonic measure is not uniform.
