The peak run lengths of evens/odds 'should' go to infinity, but these runs become a smaller and smaller component of the overall average, so that it is expected to approach the long-term 50% regardless.
In other words, an unbiased random walk should almost surely return to the origin, but a biased random walk will fail to return to the origin with nonzero probability. This can be considered a biased random walk [0], since the halting condition linearly moves further and further away from the expected value of the 50/50 walk.