Kalai's transitive-symmetric low-level Fourier-weight conjecture

For $d\geq1$, let $C_n$ be the collection of odd, transitive-symmetric functions $f:\{-1,1\}^n\to\{-1,1\}$. Let $\widehat f(S)$ denote Fourier coefficients, and let $\operatorname{Maj}_n$ be majority on $n$ bits. Kalai's transitive-symmetric low-level Fourier-weight conjecture.

$$\limsup_{n\to\infty}\sup_{f\in C_n}\sum_{|S|\leq d}\widehat f(S)^2 = \lim_{\substack{n\to\infty\\ n\text{ odd}}}\sum_{|S|\leq d}\widehat{\operatorname{Maj}_n}(S)^2.$$

This is a transitive-symmetric, odd-function version of the low-level Fourier-weight question. The source presents it as an essentially weaker conjecture attributed to Kalai and does not report a resolution.