Lemvig and Nielsen's obstruction conjecture for second-order B-spline Gabor frames

Let $Q_1(x):=\chi_{[-1/2,1/2]}(x)$ and $Q_{n+1}:=(Q_n*Q_1)$ for $n\geq 1$. For $g\in L^2(\mathbb{R})$, write $\mathcal{G}(g,a,b)=\{e^{2\pi i b m\cdot}g(\cdot-a k):k,m\in\mathbb{Z}\}$. Let

$$a_0=\dfrac{1}{2m+1},\qquad b_0=\dfrac{2k+1}{2},\qquad k,m\in\mathbb{N},\quad k>m,\quad a_0b_0<1.$$

Lemvig and Nielsen's conjecture. The Gabor system $\mathcal{G}(Q_2,a,b)$ is not a frame along the hyperbolas

$$ab=\dfrac{2k+1}{2(2m+1)},\qquad b\in\left[b_0-a_0\dfrac{k-m}{2},\,b_0+a_0\dfrac{k-m}{2}\right],$$

for every $a_0$ and $b_0$. The conjecture concerns explicit obstructions in the frame set of the second-order B-spline; the source reports that this conjecture is proved in the paper.