The extension of Theorem B for complete Padovan sequences

Let $p \geq 5$ be a prime with $\varrho_p=3$, let $\alpha$, $\beta$, and $\gamma$ be the roots of $X^3-X-1$ in $\mathbb F_p$, and define

N_p=\min\{|\alpha/\beta|,|\beta/\gamma|,|\gamma/\alpha|}.

A $\Phi_3$-sequence $(a_n)_n$ is complete when it contains every nonzero element of $\mathbb F_p$. A Padovan primitive root is a primitive root $b$ satisfying $b^3=b+1$. The extension of Theorem B. The statement of Theorem B remains true without the condition $p\leq N_p^2+1$; equivalently, a $\Phi_3$-sequence $(a_n)_n$ is complete if and only if $a_n=b^n$ for all $n$, where $b$ is a Padovan primitive root, even when $p>N_p^2+1$. The condition $p\leq N_p^2+1$ was used in the supplied theorem, while the paper reports computational and manual verification of the exceptional cases discussed there; the general extension is presented as a belief rather than proved in the supplied passage.