Conjecture on the splitting of representations in Cases 4, 7 and 8

In the cases labeled Case 4 and Cases 7 and 8, let $\pi$ denote the representation described in the surrounding case analysis. Splitting conjecture. Each such representation decomposes into two irreducible factors:

$$\pi=\pi^e+\pi^o.$$

The source states this as a conjectural decomposition, but gives no further criterion distinguishing the two factors or evidence resolving the conjecture.