Yes, this is true. Since there exists a 'decoder' $Q$ which gives a value on the right hand side (close to or equal to) $\frac{1}{2}$, the pair $(A,Q)$ are $(\ell_\infty, \ell_1)$-instance optimal (see this talk of Foucart, pg 75, for details). From this it follows that no non-trivial $2$-sparse vectors lie in the kernel of $A$ and that $A$ satisfies the null space property of order $2$ for recovery via $\ell_1$-minimization.
Chapter 11 of Foucart's book is another reference for this topic.