Friedgut--Kalai--Naor theorem for slices of the Boolean cube

The Friedgut–Kalai–Naor theorem states that if a Boolean function $f\colon {0,1}^n \to {0,1}$isclose(in$L^2$-distance)toanaffinefunction$\ell(x_1,…,x_n) = c_0 + \sum_i c_i x_i$,then$f$ is close to a Boolean affine function (which necessarily depends on at most one coordinate). We prove a similar theorem for functions defined over $\binom{[n]}{k} = {(x_1,…,x_n) \in {0,1}^n : \sum_i x_i = k }$.