Let $a,b,c,d$ be integers such that $$\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right) \mod 2$$ $$ ad-bc =1$$ $$\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \neq \left( \begin{matrix} \pm 1 & 0 \\ 0 & \pm 1 \end{matrix}\right) .$$
Let $(x,y)\in \mathbf{R}^2$ such that $-1\leq x\leq 1$ and $1/2 \leq y\leq 2$.
I highly suspect that $$c^2(y^2+2x^2) + a^2+d^2+2cx(d-a) + \frac{1}{y^2}(b-(d-a+3cx)x)^2 \geq 3.$$
The proof is actually very easy and was obtained after Parsa's answer.