For a smooth closed embedded planar curve, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus greater than 1 having the given curve as boundary, without any prescription on the conormal. By general lower bound estimates, in case the curve is a circle we prove that such problem is equivalent if restricted to embedded surfaces, we prove that do not exist minimizers, and we calculate the infimum. Then we study the case in which the genus is 1 and the competitors are restricted to a suitable class of varifolds including embedded surfaces, and we prove that the non-existence of minimizers implies a lower bound on the infimum; therefore we use such criterion in order to explicitly find an infinite family of curves for which such problem does have minimizers in the corresponding class of varifolds.