It turns out there is a (somewhat absurd) counterexample.

Consider $U=\{1,2,3,4,5,6\}$, $S_1 = \{1\}, S_2 = \{2\}, S_3 = \{3,4,5,6\}$. Then $f_{\mathbf{S}}(1) = \text{median}\{1,1,4\}=1$, yet $f_{\mathbf{S}}(2) = \text{median}\{2,5,5\}=5$.