$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp} $Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2n$-dimensional symplectic space over $F$.

Let $X,X^*$ be maximal totally isotropic subspaces of $W$, which are dual with respect to $\langle,\rangle$.

Let $e_1,\dotsc,e_n$ be a basis of $X$ and let $B_{k}$ be the parabolic subgroup of $\Sp(W)$ stabilizing the flag $$\langle e_1\rangle \subset \langle e_1, e_2\rangle \subset \dotsb \subset\langle e_1, e_2, \dotsc, e_{k}\rangle.$$

Let $\{\chi_i\}_{1\le i \le n}$ be characters of $F^{\times}$ and $N$ the unipotent subgroup of the usual Borel subgroup of $\SL_2(F)$.

Let $\pi$ be a representation of $\SL_2(F)$ whose normalized Jacquet module with respect to $N$ is $\chi_n \lvert\cdot\rvert$.

Let $\pi_1$ be an irreducible unramified representation of $\Sp(W)$ that is a subquotient of $\operatorname{Ind}_{B_{n-1}}^{\Sp(W)} (\chi_1 \otimes \dotsb \chi_{n-1} \otimes \pi)$. (Here, the induction is normalized.)

Then is $\pi_1$ a subquotient of $\operatorname{Ind}_{B_{n}}^{\Sp(W)} (\chi_1 \otimes \dotsb \chi_{n-1} \otimes \chi_n)$? I guess it should be $\operatorname{Ind}_{B_{n}}^{\Sp(W)} (\chi_1 \otimes \dotsb \chi_{n-1} \otimes \chi_n\lvert\cdot\rvert)$ since the induction and Jacquet modules are all normalized.

Any comments are welcome!