  The frame set conjecture for B-splines $B_n$, $n \ge 2$, states that the
frame set is the maximal set that avoids the known obstructions. We show that
any hyperbola of the form $ab=r$, where $r$ is a rational number smaller than
one and $a$ and $b$ denote the sampling and modulation rates, respectively, has
infinitely many pieces, located around $b=2,3,\dots$, \emph{not} belonging to
the frame set of the $n$th order B-spline. This, in turn, disproves the frame
set conjecture for B-splines. On the other hand, we uncover a new region
belonging to the frame set for B-splines $B_n$, $n \ge 2$.
