  In this paper we address a variant of the Kazhdan-Lusztig non-degeneracy
conjecture posed by Gedeon, Proudfoot and Young. We prove that if $M$ has a
free basis (something that conjecturally asymptotically all matroids are
expected to possess), then $M$ is non-degenerate. To this end, we study the
behavior of Kazhdan-Lusztig polynomials of matroids with respect to the
operation of circuit-hyperplane relaxation. This yields a family of polynomials
that relate the Kazhdan-Lusztig, the inverse Kazhdan-Lusztig and the
$Z$-polynomial of a matroid with those of its relaxations and do not depend on
the matroid. As an application of our results, we deduce that uniform matroids
maximize coefficient-wise the Kazhdan-Lusztig polynomials, inverse
Kazhdan-Lusztig polynomials and the $Z$-polynomials, when restricted to sparse
paving matroids.
