The Archimedean co-socle analogy conjecture for degenerate principal series

Let $F$ be a number field, let $P$ be a parabolic subgroup, and let $v_1$ be an Archimedean place and $v_2$ a non-Archimedean place of $F$. Write $\operatorname{I}_{P,v}(s,\chi)$ for the corresponding degenerate principal series, and let the maximal semi-simple quotient mean its co-socle. For $\operatorname{Re}(s)>0$,

Archimedean co-socle analogy conjecture. The length of the maximal semi-simple quotient of $\operatorname{I}_{P,v_1}(s,\mathbf{1})$ equals that of $\operatorname{I}_{P,v_2}(s,\mathbf{1})$. If $F_{v_1}=\mathbb{R}$, then the length of the maximal semi-simple quotient of $\operatorname{I}_{P,v_1}(s,\operatorname{sgn})$ equals that of $\operatorname{I}_{P,v_2}(s,\chi)$, where $\chi$ has order $2$.

The conjecture is intended to permit the global arguments to use spherical Archimedean sections while predicting that the relevant co-socle lengths agree with those at non-Archimedean places. No resolution status is supplied.