  Let $\pi_1,\ldots,\pi_k$ be smooth irreducible representations of $p$-adic
general linear groups. We prove that the parabolic induction product
$\pi_1\times\cdots\times \pi_k$ has a unique irreducible quotient whose
Langlands parameter is the sum of the parameters of all factors (cyclicity
property), assuming that the same property holds for each of the products
$\pi_i\times \pi_j$ ($i<j$), and that for all but at most two representations
$\pi_i\times \pi_i$ remains irreducible (square-irreducibility property). Our
technique applies the recently devised Kashiwara-Kim notion of a normal
sequence of modules for quiver Hecke algebras.
  Thus, a general cyclicity problem is reduced to the recent Lapid-M\'inguez
conjectures on the maximal parabolic case.
