Nonzero-determinant conjecture for the Suzuki-group recognition equations

Let $P,Q\in\mathcal{O}$, let $h\in G$, and let $x\in\operatorname{GL}(4,q)$. Define

$$P^{\prime}=Px^{-1},\qquad Q^{\prime}=Qx^{-1},\qquad g=h^{x^{-1}}.$$

Regard the four equations in as simultaneous linear equations in the variables $\alpha^{nt}$ for $n=1,\dotsc,4$, over the polynomial ring $\mathbb{F}_q[\alpha]$. Nonzero-determinant conjecture. For every such choice of $P$, $Q$, $h$, and $x$, this linear system has non-zero determinant. This conjecture would provide the single polynomial in $\alpha$ of bounded degree needed in the recognition argument, by eliminating the variables $\alpha^{nt}$ from the system.