The Poisson avoidance conjecture for a system of linear forms

Let $q$ be a prime, let $\mathbb F_q$ be the finite field with $q$ elements, let $d=\pi(q-1)$, and let $L_n$ be the linear forms from the paper on $\mathbb F_q^d$. For $x_0\in\mathbb F_q$, the Poisson avoidance conjecture. As $q\to\infty$,

$$\frac{\#\{\mathbf v\in\mathbb F_q^d: L_n(\mathbf v)\neq x_0\text{ for }1\leq n\leq q\}}{q^d}=e^{-1}+o(1).$$

The assertion expresses the expected Poisson probability that none of the $q$ prescribed linear conditions is met. The supplied passage gives the expected asymptotic but does not indicate a proof or resolution.