Suppose that $Y^n$ is obtained by observing a uniform Bernoulli random vector $X^n$ through a binary symmetric channel with crossover probability $alpha$. The ``most informative Boolean function'' conjecture postulates that the maximal mutual information between $Y^n$ and any Boolean function $b(X^n)$ is attained by a dictator function. In this paper, we consider the ``complementary" case in which the Boolean function is replaced by $f:left{0,1right}^nmapstoleft{0,1right}^{n-1}$, namely, an $n-1$ bit quantizer, and show that $I(f(X^n);Y^n)leq (n-1)cdotleft(1-h(alpha)right)$ for any such $f$. Thus, in this case, the optimal function is of the form $f(x^n)=(x_1,ldots,x_{n-1})$.