  This is a continuation of a series of papers studying the birational Mori
fiber structures of anticanonically embedded $\mathbb{Q}$-Fano $3$-fold
weighted complete intersections of codimension $2$. We have proved that $19$
families consists of birationally rigid varieties and $14$ families consists of
birationally birigid varieties. The aim of this paper is to continue the work
systematically and prove that, among the remaining $52$ families, $21$ families
consist of birationally birigid varieties.
