Symmetry of the counting function in a unitary polar-space construction

Let $q$ be an odd prime power, let $b\in\mathbb{F}_{q^2}$ satisfy $b^{q+1}=-1$, and let $(s,u,w)\in\mathbb{F}_{q^2}^3$ satisfy

$$s^{q+1}=u^{q+1}+w^{q+1}.$$

For each $\kappa\in\mathbb{F}_q^*$, let $n(\kappa)$ be the number of values of $\lambda\in\mathbb{F}_{q^2}\cup\{\infty\}$ satisfying

$$(w\lambda+1)(b+u\lambda)^q-(w\lambda+1)^q(b+u\lambda)=\kappa\left(b^2+1+\lambda^2(s^2-u^2-w^2)\right)^{(q+1)/2}.$$

Counting-function symmetry conjecture. For every $\kappa\in\mathbb{F}_q^*$, one has

$$n(\kappa)=n(-\kappa).$$

This conjecture is introduced to support a construction concerning unitary groups acting on totally isotropic $k$-spaces. The source does not report a resolution, so the conjecture remains open.