  In this note we define the notion of collarable slices of Lagrangian
submanifolds. Those are slices of Lagrangian submanifolds which can be isotoped
through Lagrangian submanifolds to a cylinder over a Legendrian embedding near
a contact hypersurface. Such a notion arises naturally when studying
intersections of Lagrangian submanifolds with contact hypersurfaces. We then
give two explicit examples of Lagrangian disks in $\mathbb{C}^2$ transverse to
$S^3$ whose slices are non-collarable.
