The low-rank reduction conjecture for Jordan–Hölder factors of real principal series

Let $Ps(\underline{\pi})=\pi_1\times\cdots\times\pi_t$ be a standard module of $\operatorname{GL}_n({\mathbb R})$, where each $\pi_i$ is an essentially discrete series representation of $\operatorname{GL}_{n_i}({\mathbb R})$, with $n_i\leq 2$, and Langlands parameter $\sigma_i$. Set

$$\tau_0=\sigma_1+\sigma_2+\cdots+\sigma_t.$$

Low-rank reduction conjecture. An irreducible admissible representation $\pi$ of $\operatorname{GL}_n({\mathbb R})$ occurs as a Jordan–Hölder factor of $Ps(\underline{\pi})$ if and only if its Langlands parameter $\sigma_\pi$ can be obtained from $\tau_0$ by a sequence $\tau_0,\tau_1,\ldots,\tau_r$, where each $\tau_{i+1}$ is obtained from $\tau_i$ by replacing a summand $\sigma$ with a summand $\sigma'$ satisfying

$$\dim\sigma=\dim\sigma'\leq 4,\qquad \sigma'\geq\sigma.$$

Thus every Jordan–Hölder factor should be detectable by analyzing groups $\operatorname{GL}_m({\mathbb R})$ with $m\leq4$. The conjecture is motivated by the explicit but unresolved decomposition in rank four; the source gives no proof or resolution.