Let $X$ be a scheme with a point $x \in X$ and $\mathcal E$ a locally free sheaf of finite rank on $X$ with a global section $s \in \mathcal E(X)$.
If $\varphi \in Hom_{\mathcal O_X}(\mathcal E, \mathcal O_X)$, then $\varphi(X)(s) \in \mathcal O_X(X)$.
Suppose that the class of $\varphi(X)(s)$ in $\mathcal O_{X,x}/\mathcal m_x$ is zero for all $\varphi \in Hom_{\mathcal O_X}(\mathcal E, \mathcal O_X)$.
Does this imply that the class of $s$ in $\mathcal E_x /\mathcal m_x \cdot \mathcal E_x$ is zero?