Non-factorability conjecture for the character on smooth polynomial congruences

Let $n$, $m$, and $\alpha$ and the notation above be given. Let $\mathcal{P}_S$ be the set of smooth elements of the multiplicative semigroup $\mathcal{P}$, and let $\chi_{\mathcal{P}}$ be the associated degree-one order-two character. The characters $\chi_{\mathfrak{p}}$ and $(-1)^{\operatorname{ord}_p(P(m))}$, $(-1)^{\operatorname{ord}_{\mathfrak{p}}(P(\alpha))}$ are the characters described above. Non-factorability conjecture. The restriction of $\chi_{\mathcal{P}}$ to $\mathcal{P}_S$ cannot be written as a product of the characters $\chi_{\mathfrak{p}}$, $(-1)^{\operatorname{ord}_p(P(m))}$, and (-1)^{\operatorname{ord}_{\mathfrak{p}}(P(\alpha)). This asserts that the character governing the found congruences has a genuinely nontrivial component beyond the specified local character factors; the supplied text does not give evidence resolving the claim.