The multiplicative generation conjecture for generalized Motzkin numbers

Let $a,b\in\mathbb{Z}$, let $p$ be a prime, and define

$$S_p=\left\{T^{a,b}_n\mid n\in\mathbb{F}_p\right\}.$$

Assume that $p$ does not divide any member of $S_p$. Multiplicative generation conjecture. The set $S_p$ generates $\mathbb{F}_p^\times$ as a multiplicative group. Equivalently,

$$\left\{T^{a,b}_n\bmod p\mid n\in\mathbb{N}\right\}=\mathbb{F}_p^\times.$$

Consequently, all values of $\mathbb{F}_p^\times$ appear in $T^{a,b}_n$ and $M^{a,b}_n$ modulo $p$, with equal density. The claim concerns the distribution and symmetry of generalized Motzkin numbers modulo primes; the supplied text does not establish its resolution, so its status remains open.