  We consider two instances of boundary conditions for massless scalars on
$AdS_2$ that interpolate between the Dirichlet and Neumann cases while
preserving scale invariance. Assessing invariance under the full
$SL(2;\mathds{R})$ 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 $AdS_2\subset
AdS_4\times\mathbb{CP}^3$ worldsheet, thus obtaining correlators of operators
inserted along a $1$-dimensional defect in ${\cal N}=6$ 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.
