Menezes–Teske–Weng's finite-field quadratic conjecture

Let $l\in\mathbb{N}$, set $q=2^l$, and let $\beta\in\mathbb{F}_{q^6}^*$. Suppose that the quadratic equation

$$u^2+(\beta^{q^4-1}+\beta^{q^2-1}+1)\nu+\beta^{q^2-1}=0$$

has two solutions $u_1,u_2\in\mathbb{F}_{q^6}$. Menezes–Teske–Weng's conjecture. The solutions satisfy

$$u_i^{q^2+1}+u_i+1=0\qquad (i=1,2).$$

This conjecture concerns the algebraic condition used in a proposed reduction of discrete logarithms on certain elliptic curves over characteristic-two finite fields to a divisor-class-group discrete logarithm problem. Its resolution status is not specified in the supplied text.