Conjectural Gauss-sum formula for the Walsh transform of binomial Boolean functions

For $nu=2$, let $a\in\mathbb{F}_{2^{m_1}}^*$ satisfy $K_{m_1}(a)\equiv1\pmod 3$, let $b\in\mathbb{F}_4^*$, and let $\omega\in\mathbb{F}_{2^{m_0}}^*$. Define $\gamma=b\psi_{m_1}(w_2)$, where $\psi_{m_1}$ is the cubic multiplicative character, and let $f_a$ and $S_2(a,b,\omega)$ be the functions and sum defined in the source.

Conjectural Gauss-sum formula. There exists a Boolean function $h_{a,b}(\omega)$ such that

$$S_2(a,b,\omega)=\frac{2^{m_2+1}-K_{m_1}(a)}{3}-2f_a(w_1^{-1})\frac{2^{m_2+1}-1}{3}-h_{a,b}(\omega)\chi\left(f_a(w_1^{-1})\right)2^{m_2},$$

and, for $\omega\ne0$,

$$\widehat{\chi_{f_{a,b}}}(\omega)=\chi\left(h_{a,b}(\omega)f_a(w_1^{-1})\right)2^{m_1}+\frac{4-K_{m_1}(a)}{3}.$$

The source reports experimental evidence for this relation because the preceding methods do not handle the general case $w_1\ne1$. The formula would relate the general Walsh transform to the subfield case and is intended to support the bentness criterion; no proof is supplied.