  For a smooth closed embedded planar curve $\Gamma$, we consider the
minimization problem of the Willmore energy among immersed surfaces of a given
genus $\mathfrak{g}\geq1$ having the curve $\Gamma$ as boundary, without any
prescription on the conormal. By general lower bound estimates, in case
$\Gamma$ is a circle we prove that such problem is equivalent if restricted to
embedded surfaces, we prove that do not exist minimizers, and the infimum
equals $\beta_\mathfrak{g}-4\pi$, where $\beta_\mathfrak{g}$ is the energy of
the closed minimizing surface of genus $\mathfrak{g}$. We also prove that the
same result also holds if $\Gamma$ is a straight line for the suitable
analogously defined minimization problem on asymptotically flat surfaces.\\
Then we study the case in which $\Gamma$ is compact, $\mathfrak{g}=1$ and the
competitors are restricted to a suitable class $\mathcal{C}$ of varifolds
including embedded surfaces. We prove that under suitable assumptions
minimizers exists in this class of generalized surfaces.
