  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 $k \times k$ and smaller are positive). We show that the
Kazhdan-Lusztig immanant indexed by $v$ is positive on $k$-positive matrices
when $v$ avoids 1324 and 2143 and for all non-inversions $i<j$ of $v$, either
$j-i \leq k$ or $v_j-v_i\leq k$. Our main tool is Lewis Carroll's identity.
