Parabolic induction from two segments, linked under contragredient, with a one half cuspidal reducibility, a special case

In this paper, we determine the composition series of the induced representation $\delta([\nu^{-a}\rho,\nu^{c}\rho]) \times \delta([\nu^{\frac{1}{2}}\rho, \nu^{b}\rho]) \rtimes \sigma$ where $a, b, c \in \mathbb{Z}+ \frac{1}{2}$ such that $\frac{1}{2} \leq a < b < c$, $\rho$ is an irreducible cuspidal unitary representation of a general linear group and $\sigma$ is an irreducible cuspidal representation of a classical group.