  The Cabling Conjecture of Gonz\'alez-Acu\~na and Short holds that only cable
knots admit Dehn surgery to a manifold containing an essential sphere. We
approach this conjecture for thin knots using Heegaard Floer homology,
primarily via immersed curves techniques inspired by Hanselman's work on the
Cosmetic Surgery Conjecture. We show that almost all thin knots satisfy the
Cabling Conjecture, with possible exception coming from a (conjecturally
non-existent) collection of thin, hyperbolic, L-space knots. This result serves
as a reproof that the Cabling Conjecture is satisfied by alternating knots, and
also a new proof that thin, slice knots satisfy the Cabling Conjecture.
