Approximate spectral norm conjecture for symmetric Boolean functions

For a Boolean function $f:\{0,1\}^n\to\{\pm1\}$, let $\|\widehat f\|_{1,\epsilon}$ be the smallest spectral norm of a function $g$ satisfying

$$\|f-g\|_\infty\leq\epsilon.$$

The function $f$ is symmetric when its value depends only on the Hamming weight of the input. Approximate spectral norm conjecture. For every symmetric function $f$,

$$\log \|\widehat f\|_1=\Theta^*(\log \|\widehat f\|_{1,1/3}),$$

where $\Theta^*$ suppresses $O(\log n)$ factors. Extending the spectral-norm theorem to approximate spectral norm would have implications for learning and complexity measures; the conjectured comparison is presented as an open next step.