  For a class of compactly supported windows we characterize the frame property
for a Gabor system $\mts,$ for translation parameters $a$ belonging to a
certain range depending on the support size. We show that the obstructions to
the frame property are located on a countable number of "curves." For functions
that are positive on the interior of the support these obstructions do not
appear, and the considered region in the $(a,b)$ plane is fully contained in
the frame set. In particular this confirms a recent conjecture about B-splines
by Gr\"ochenig in that particular region. We prove that the full conjecture is
true if it can be proved in a certain "hyperbolic strip."
