Bobkov and Houdre https://projecteuclid.org/euclid.bj/1178291721 showed that among all $f:R^n\to R$ that are $1$-Lipschitz with respect to the $\ell_1$ metric, the variance is maximized by sums. Namely, if $\xi_i$ are independent with variance $\sigma_i^2=\mathrm{var}[\xi_i]$, then $$ \mathrm{var}[ f(\xi_1,\ldots,\xi_n)]\le \sum_{i=1}^n\sigma_i^2. $$
Motivated by this result, we ask the following question. Let $\xi=(\xi_1,\ldots,\xi_n)\in\{0,1\}^n$ have joint distribution $\mu$. Suppose that $\mu$ satisfies the following property: for all functions $f:\{0,1\}^n\to R$ of the form $$ f(x)=\sum_{i=1}^n f_i(x_i), $$ where $f_i:\{0,1\}\to[0,1]$, we have $$ P( |f(\xi)-E f(\xi)|\ge t) \le 2\exp(-Ct^2), \qquad t\ge0, \quad(*) $$ where $P$ and $E$ are defined with respect to $\mu$ and $C$ depends on $\mu$ only. For example, product measures satisfy $(*)$ with $C=2/n$.
Question: does $(*)$ necessarily also hold for all $1$-Lipschitz $f:\{0,1\}^n\to R$ (under the Hamming metric)?
