Obstructions for Gabor frames of the second order B-spline

For a window $g\in L^2(\mathbb{R})$, the subset of all lattice parameters $(a, b)\in \mathbb{R}^2_+$ such that $\mathcal{G}(g,a,b)=\{e^{2\pi ib m\cdot}g(\cdot-a k) : k, m\in\mathbb{Z}\}$ forms a frame for $L^2(\mathbb{R})$ is known as the frame set of $g$. In time-frequency analysis, determining the Gabor frame set for a given window is a challenging open problem. In particular, the frame set for B-splines has many obstructions. Lemvig and Nielsen in \cite{counter} conjectured that if \begin{align} a_0=\dfrac{1}{2m+1},~ b_0=\dfrac{2k+1}{2},~k,m\in \mathbb{N},~k>m,~a_0b_0<1,\nonumber \end{align} then the Gabor system $\mathcal{G}(Q_2, a, b)$ of the second order B-spline $Q_2$ is not a frame along the hyperbolas \begin{align} ab=\dfrac{2k+1}{2(2m+1)},\text{ for }b\in \left[b_0-a_0\dfrac{k-m}{2}, b_0+a_0\dfrac{k-m}{2}\right],\nonumber \end{align} for every $a_0$, $b_0$. Nielsen in \cite {Nielsenthesis} also conjectured that $\mathcal{G}(Q_2, a,b)$ is not a frame for $$a=\dfrac{1}{2m},~b=\dfrac{2k+1}{2},~k,m\in \mathbb{N},~k>m,~ab<1\text{ with }\gcd(4m,2k+1)=1.$$ In this paper, we prove that both conjectures are true.