Connection between heights of $(U:_R E)$ and $(ann(E):_R ann(U))$ for torsion-free modules $U\subseteq E$ of constant rank

Question

Let $R$ be a Noetherian local ring. Let $E$ be a finitely generated torsion-free $R$-module of constant rank $e$. Let $s$ be an integer such that $s\geq e+1$. Let $U$ be an $R$-submodule of $E$ and set $I :=(U:_R E)(=ann(E/U))$ and $J :=(ann(E):_R ann(U))$.
My very broad question is: Is there any connection between $ht(I)$ and $ht(J)$? Very specifically, is there any implication(s) between the following two statements:

(1) $ht(J)\geq s-e$.

(2) $ht(I) \geq s-e$.

?