  In this note, we show that certain dual canonical basis elements of
$\mathbb{C}[SL_m]$ are positive when evaluated on $k$-positive matrices,
matrices whose minors of size $k \times k$ and smaller are positive. Skandera
showed that all dual canonical basis elements of $\mathbb{C}[SL_m]$ can be
written in terms of Kazhdan-Lusztig immanants, which were introduced by Rhoades
and Skandera. We focus on the basis elements which are expressed in terms of
Kazhdan-Lusztig immanants indexed by 1324- and 2143-avoiding permutations. This
extends previous work of the authors on Kazhdan-Lusztig immanants and uses
similar tools, namely Lewis Carroll's identity (also known as the
Desnanot-Jacobi identity).
