Dobbertin's conjecture on the linearity of monomial vectorial functions

Let $m$ and $n$ be the parameters of the vectorial function $F$ under consideration, and let $u$ satisfy

$$\gcd(u,2^m-1)=\gcd(u-2^e,2^m-1)=1,$$

with $u\not\equiv 2^k\pmod{2^m-1}$ for every $k\geq 0$. Define

$$\mathcal{L}(u)=\max_{\omega\in\mathbb{F}_{2^m}}|W_{H_1}(\omega)|,$$

where $H(x)=x^u$. Dobbertin's conjecture. The linearity satisfies

$$\mathcal{L}(u)\geq 2^{\left\lfloor\frac{n}{4}+1\right\rfloor},$$

equivalently, the nonlinearity of $F$ satisfies

$$\mathcal{N}_F\leq 2^{n-1}-2^{\left\lfloor\frac{3n}{4}\right\rfloor}.$$

This conjecture gives a lower bound for the Walsh-spectrum maximum of the monomial permutation and, through the stated relation between $\mathcal{L}(u)$ and $\mathcal{N}_F$, an upper bound on the nonlinearity of the associated vectorial function. The supplied text does not state whether the conjecture has been proved or disproved.