  In this note, we prove that the problem of computing the linear discrepancy
of a given matrix is $\Pi_2$-hard, even to approximate within $9/8 - \epsilon$
factor for any $\epsilon > 0$. This strengthens the NP-hardness result of Li
and Nikolov [ESA 2020] for the exact version of the problem, and answers a
question posed by them. Furthermore, since Li and Nikolov showed that the
problem is contained in $\Pi_2$, our result makes linear discrepancy another
natural problem that is $\Pi_2$-complete (to approximate).
