We consider two instances of boundary conditions for massless scalars on AdS2 that interpolate between the Dirichlet and Neumann cases while preserving scale invariance. Assessing invariance under the full SL(2; ℝ) conformal group is not immediate given their non-local nature. To further clarify this issue, we compute holographically 2- and 4-point correlation functions using the aforementioned boundary conditions and study their transformation properties. Concretely, motivated by the dual description of some multi-parametric families of Wilson loops in ABJM theory, we look at the excitations of an open string around an AdS2 ⊂ AdS4 × ℂℙ3 worldsheet, thus obtaining correlators of operators inserted along a 1-dimensional defect in N = 6 super Chern-Simons-matter theory at strong coupling. Of the two types of boundary conditions analyzed, only one leads to the expected functional structure for conformal primaries; the other exhibits covariance under translations and rescalings but not under special conformal transformations.