The $T$-adic Galois representation is surjective for a positive density of Drinfeld modules

Let ${\mathbb{F}}_q$ be the finite field with $q\ge 5$ elements, $A:={\mathbb{F}}_q[T]$ and $F:={\mathbb{F}}_q(T)$. Assume that $q$ is odd and take $\mathrm{\mid }\cdot \mathrm{\mid }$ to be the absolute value at $\mathrm{\infty }$ that is normalized by $\mathrm{\mid }T\mathrm{\mid }=q$. Given a pair $w=(g_1,g_2)\in A^2$ with $g_2\mathrm{\ne }0$, consider the associated Drinfeld module ${\phi }^w:A\to A\{\tau \}$ of rank $2$ defined by ${\phi }_T^w=T+g_1\tau +g_2{\tau }^2$. Fix integers $c_1,c_2\ge 1$ and define $\mathrm{\mid }w\mathrm{\mid }:=max\{\mathrm{\mid }{g}_1{\mathrm{\mid }}^{\frac{1}{c_1}},\mathrm{\mid }{g}_2{\mathrm{\mid }}^{\frac{1}{c_2}}\}$. I show that when ordered by height, there is a positive density of pairs $w=(g_1,g_2)$, such that the $T$-adic Galois representation attached to ${\phi }^w$ is surjective.