Order of torsion for reduction of linearly independent points for a family of Drinfeld modules

Let $q$ be a power of the prime number $p$, let $K={\mathbb{F}}_q(t)$, and let $r\ge 2$ be an integer. For points $\mathbf{a},\mathbf{b}\in K$ which are ${\mathbb{F}}_q$-linearly independent, we show that there exist positive constants $N_0$ and $c_0$ such that for each integer $\mathrm{\ell }\ge N_0$ and for each generator $\tau$ of ${\mathbb{F}}_{q^{\mathrm{\ell }}}\mathrm{/}{\mathbb{F}}_q$, we have that for all except $N_0$ values $\lambda \in \overline{{\mathbb{F}}_q}$, the corresponding specializations $\mathbf{a},\mathbf{b}(\tau )$ and $\mathbf{b}(\tau )$ cannot have orders of degrees less than $c_0\log \log \mathrm{\ell }$ as torsion points for the Drinfeld module $\Phi^{(\tau,\lambda)}:$\mathbb{F}$q[T] {\longrightarrow} {$\mathrm{E}\mathrm{n}\mathrm{d}$}{\overline{$\mathbb{F}$_q}}({\mathbb G}_a)$(where${\mathbb G}_a$istheadditivegroupscheme),givenby$\Phi^{(\tau,\lambda)}_T(x)=\tau x+\lambda x^q + x^{q^r}$.