The non-frame conjecture for singular counterexamples to the B-spline Gabor frame property

Let $\mathcal{G}(B_2,a,b)$ denote the Gabor system generated by the quadratic B-spline $B_2$ with parameters $a,b>0$. Let $k,m\in\mathbb{N}$ satisfy $k>m$ and $a_0b_0<1$, where

$$a_0=\frac{1}{2m+1},\qquad b_0=\frac{2k+1}{2}.$$

Non-frame conjecture. The Gabor system $\mathcal{G}(B_2,a_0,b_0)$ is not a frame. Furthermore, it is not a frame along the hyperbolas

$$ab=\frac{2k+1}{2(2m+1)},$$

for

$$b\in\left[b_0-a_0\frac{k-m}{2},\,b_0+a_0\frac{k-m}{2}\right],$$

for every $a_0$ and $b_0$ defined above.

This conjecture proposes that the counterexamples found at isolated parameter pairs extend along specified hyperbolic segments. The supplied text does not establish whether this conjecture has been resolved.