Let $\chi_X:\{-1,1\}^n\to \{0,1\}$ be the characteristic function of a subset $X\subseteq \{-1,1\}^n$, which is randomly drawn from all subsets with exactly $k$ elements.

Is the support of the Fourier transform (Walsh-Hadamard transform) $\hat{\chi}_X$ large ($\geq c2^n$ for a constant $c>0$) with high probability?