Adduced representation formula for unitary representations of $GL(n,\mathbb{R})$

Let $G_n=GL(n,\mathbb{R})$, let $\pi\in\widehat{G_n}$, and write $\pi=\pi_1\times\cdots\times\pi_k$ according to the Vogan classification, with each $\pi_i$ a basic unitary representation of one of the listed types. For a basic representation, define the adduced representation $A\pi_i$ by lowering its segment parameter as follows:

$$\begin{aligned} A\left(\chi\left(m,\varepsilon,it\right)\right)&=\chi\left(m-1,\varepsilon,it\right),\\ A\left(\sigma\left(2m,s;\varepsilon,it\right)\right)&=\sigma\left(2\left(m-1\right),s;\varepsilon,it\right),\\ A\left(\delta\left(2m,k;it\right)\right)&=\delta\left(2\left(m-1\right),k;it\right),\\ A\left(\psi\left(4m,k,s;it\right)\right)&=\psi\left(4\left(m-1\right),k,s;it\right). \end{aligned}$$

Adduced representation conjecture. The adduced representation of $\pi$ is the parabolic product of the adduced representations of its basic factors:

$$A\pi=A\pi_1\times\cdots\times A\pi_k.$$

This formula describes how the adduced representation operation interacts with the Vogan classification of irreducible unitary representations of $GL(n,\mathbb{R})$. The supplied status evidence says that the corresponding result is proved in the cited work, so the claim is recorded as solved.