On the independence of Heegner points on elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$ with no CM and $\mathcal{O}_{1},\ldots,\mathcal{O}_{r}$ be orders in distinct imaginary quadratic fields $k_{1},\ldots,k_{r}$, respectively, which satisfy the Heegner condition for $N$. Let $P_{1},\ldots,P_{r}$ be the Heegner points on $E$ attached to $\mathcal{O}_{1},\ldots,\mathcal{O}_{r}$, respectively. Silverman and Rosen proved that if $Cond(\mathcal{O}_{1})=\dots=Cond(\mathcal{O}_{r})=1 $, then there is a constant $C$ such that if, for each $i$, $\#{Pic(\mathcal{O}_{i})}^{odd}\ge{C}$ then the points $P_{1},\ldots,P_{r}$ are independent in $E(\overline{\mathbb{Q}})/E_{tors}(\overline{\mathbb{Q}})$. In this paper we show that the condition $Cond(\mathcal{O}_{1})=\dots=Cond(\mathcal{O}_{r})=1 $ .is not necessary and it can be removed