Notation: identify an element of $\{-1,1\}^n$ with the set $S \subseteq \{1, \ldots, n\}$ on which it takes the value $-1$.

The following is an asymptotic question. "Close to one" means "more than $r_n$" and "away from $\frac{n}{2}$" means "outside the interval $[\frac{n}{2} - k_n\sqrt{n}, \frac{n}{2} + k_n\sqrt{n}]$", for some $r_n$ and $k_n$ which respectively increase to one and infinity as $n \to \infty$.

Conjecture: for any subset of $\{-1,1\}^n$ there is a complex valued function with $l^2$ norm equal to 1, supported either on that subset or on its complement, whose Fourier transform has $l^2$ norm close to one on $\{S: |S|$ is away from $\frac{n}{2}\}$.

A positive answer would have very interesting consequences. It would mean that a single subspace of $l^2(\{-1,1\}^n)$ whose dimension is small compared to the whole space is close to every subspace spanned by standard basis vectors or its complement.

I'm not familiar with the literature on Fourier analysis on the discrete cube. Is there anything there that would help settle this question?

Edit: I would have edited this earlier, but I assumed it had disappeared from view. Since the Kadison-Singer problem has a positive solution, the answer to the conjecture is no. Posting details in an answer.