Abstract: Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan-Lusztig immanants, which are indexed by permutations, involve q=1 specializations of Type A Kazhdan-Lusztig polynomials, and were defined in (Rhoades-Skandera, 2006). Using results of (Haiman, 1993) and (Stembridge, 1991), Rhoades and Skandera showed that Kazhdan-Lusztig immanants are nonnegative on matrices whose minors are nonnegative. We investigate which Kazhdan-Lusztig immanants are positive on k-positive matrices (matrices whose minors of size k×k and smaller are positive). For v a permutation avoiding 1324 and 2143, we give a sufficient condition on k so that the Kazhdan-Lusztig immanant indexed by v is positive on k-positive matrices. Our main tool is the Desnanot-Jacobi identity.