  We prove a lower bound of $\Omega(n^{1/2 - c})$, for all $c>0$, on the query
complexity of (two-sided error) non-adaptive algorithms for testing whether an
$n$-variable Boolean function is monotone versus constant-far from monotone.
This improves a $\tilde{\Omega}(n^{1/5})$ lower bound for the same problem that
was recently given in [CST14] and is very close to $\Omega(n^{1/2})$, which we
conjecture is the optimal lower bound for this model.
