Condition for a curve to bound disks on the surface of a handlebody

Question

Let $\Sigma$ be a closed orientable surface of genus $g$ and let $\alpha = \{\alpha_1,...,\alpha_g\}$ be a set of pairwise nonintertersecting nonseparating simple closed curves on $\Sigma$. Then $\alpha$ determines a way of identifying the boundary of a genus $g$ handlebody $H$ with $\Sigma$. If $\gamma$ is a simple closed curve in $\Sigma$ that does not intersect any of the $\alpha_i$, does it follow that $\gamma$ will bound a properly embedded disk when we identify $\Sigma$ with $\partial H$?