Reinhardt domains with a cusp at the origin

Let V be a bounded pseudoconvex Reinhardt domain in C^2 with many strictly pseudoconvex points and logarithmic image W. It was known that the maximal ideal in $H^{\infty}(V)$ consisting of all functions vanishing at (p,q) in V is generated by the coordinate functions z-p, w-q (meaning that one can solve the Gleason problem for $H^{\infty}(V)$) if W is bounded. We show that one can solve Gleason's problem for $H^{\infty}(V)$ as well if there are positive numbers $a$, $b$ and a positive rational number k/l such that V looks like {(z,w) in C^2 : a |w|^l <= |z|^k = b |w|^l} for small (z,w).