Optimal Boolean Locality-Sensitive Hashing

For $0 \leq \beta < \alpha < 1$ the distribution $\mathcal{H}$ over Boolean functions $h \colon \{-1, 1\}^d \to \{-1, 1\}$ that minimizes the expression \begin{equation*} \rho_{\alpha, \beta} = \frac{\log(1/\Pr_{\substack{h \sim \mathcal{H} \\ (x, y) \text{ $\alpha$-corr.}}}[h(x) = h(y)])}{\log(1/\Pr_{\substack{h \sim \mathcal{H} \\ (x, y) \text{ $\beta$-corr.}}}[h(x) = h(y)])} \end{equation*} assigns nonzero probability only to members of the set of dictator functions $h(x) = \pm x_i$.