Boolean function monotonicity testing requires (almost) $n^{1\mathrm{/}2}$ queries

We show that for any constant $c>0$, any (two-sided error) adaptive algorithm for testing monotonicity of Boolean functions must have query complexity $\mathrm{\Omega }(n^{1\mathrm{/}2-c})$. This improves the $\tilde{\mathrm{\Omega }}(n^{1\mathrm{/}3})$ lower bound of [CWX17] and almost matches the $\tilde{O}(\sqrt{n})$ upper bound of [KMS18].