Bentness criterion for binomial Boolean functions in degree four times an odd number

Let $n=4m$ with $m$ odd, let $a\in\mathbb{F}_{2^{n/2}}^*$ and $b\in\mathbb{F}_4^*$, and let $f_{a,b}$ be the binomial Boolean function defined by

$$f_{a,b}(x)=\operatorname{Tr}_{1}^{n}\left(a x^{r(2^{n/2}-1)}\right)+\operatorname{Tr}_{1}^{2}\left(bx^{(2^n-1)/3}\right).$$

Bentness conjecture. The function $f_{a,b}$ is bent if and only if $K_{n/2}(a)=4$.

The conjecture is motivated by an explicit Walsh-transform formula on the subfield and extensive experimental evidence for arbitrary $\omega\in\mathbb{F}_{2^n}$; proving the formula would establish this criterion. The source presents the claim as open and suggests analogous results for other $2$-adic valuations of the extension degree.