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 $(\ell_2, \ell_1)$-instance optimal (see [this talk][1] 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][2] is another reference for this topic.

  [1]: http://math.univ-lille1.fr/~cempi/activites_scientifiques/FR/conf/files/Lecon3B.pdf
  [2]: https://link-springer-com.prx.library.gatech.edu/book/10.1007%2F978-0-8176-4948-7