The surjectivity characterization conjecture for LR-separated cellular automata

Let $F$ be a $LR$-separated cellular automaton over the finite field $\mathbb Z_{p}$, where $p\geq 3$ is prime, and let $\ell$ (respectively, $r$) be the leftmost (respectively, rightmost) position of $F$.

Surjectivity characterization conjecture. $F$ is surjective if and only if either

$$\gcd(q_{\ell},p-1)=1$$

or

$$\gcd(q_r,p-1)=1.$$

The conjecture proposes a complete characterization of surjective $LR$-separated cellular automata over finite fields, extending the one-sided sufficient criterion established for finite rings. The converse fails over general finite rings, so the finite-field hypothesis is essential to the proposed statement; its resolution is not given here.