Let $\Sigma$ be a compact orientable connected $2$-manifold with a non-empty boundary. Let $\widehat \pi(\Sigma)$ denote the set of free homotopy classes of curves in $\Sigma$. We say $x\in \widehat \pi(\Sigma)$ is peripheral, if there is representative $\alpha\colon \Bbb S^1\to \Sigma$ of $x$ with $\text{im}(\alpha)\subseteq \partial \Sigma$.

Question:Let $x\in\widehat \pi(\Sigma)$ be anon-trivial non-peripheralelement. Does there exist a simple closed curve $\beta\subseteq \Sigma$ such that the geometric intersection of $\alpha$ and $\beta$ is non-zero, where $\alpha$ is a representative of $x$.