Let L be a closed manifold of dimension n ≥ 2 which admits a totally real embedding into C^n. Let ST*L be the space of rays of the cotangent bundle T*L of L and let DT*L be the unit disc bundle of T*L defined by any Riemannian metric on L. We observe that ST*L endowed with its standard contact structure admits weak symplectic fillings W which are diffeomorphic to DT*L and for which any closed Lagrangian submanifold N ⊂ W has the property that the map H_1(N, R) → H_1(W, R) has a nontrivial kernel. This relies on a variation on a theorem by Laudenbach and Sikorav.