In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6
′′, p. 325):

Let $K$ be a number field and let 
$K^{cycl}$ be the cyclotomic extension of 
$K$ generated by all roots of unity. 
Let $E$ and $E'$ be two elliptic curves such that, over $\bar{K}$,

(i) $E$ and $E'$ have no complex multiplication;

(ii) The $l$-adic representations $(\rho_l)$, $(\rho'_l)$ attached to $E$ and $E'$ don't become isomorphic over any finite extension of $K$.

Then $K(E_{tors}) \cap K(E'_{tors})$ is finite over $K^{cycl}$.

My question is whether this holds for $E$ and $E'$ defined over a function field?