  Let $n=2m$. In the present paper, we study the binomial Boolean functions of
the form $$f_{a,b}(x) = \mathrm{Tr}_1^{n}(a x^{2^m-1 })
+\mathrm{Tr}_1^{2}(bx^{\frac{2^n-1}{3} }), $$ where $m$ is an even positive
integer, $a\in \mathbb{F}_{2^n}^*$ and $b\in \mathbb{F}_4^*$. We show that $
f_{a,b}$ is a bent function if the Kloosterman sum
$$K_{m}\left(a^{2^m+1}\right)=1+ \sum_{x\in \mathbb{F}_{2^m}^*}
(-1)^{\mathrm{Tr}_1^{m}(a^{2^m+1} x+ \frac{1}{x})}$$ equals $4$, thus settling
an open problem of Mesnager. The proof employs tools including computing Walsh
coefficients of Boolean functions via multiplicative characters, divisibility
properties of Gauss sums, and graph theory.
