Divisibility of the characteristic polynomial by a common divisor

Let $A=\mathbb{K}[p,q]$ be the subalgebra generated by $p$ and $q$, let $\chi_{p,q}(x)$ be their characteristic polynomial, and let $a(x)$ be a polynomial of degree at least two dividing both $p(x)$ and $q(x)$. Divisibility conjecture. It is conjectured that

$$a(x)\mid\chi_{p,q}(x).$$

The preceding theorem shows that every root of $a(x)$ is a root of $\chi_{p,q}(x)$; the conjecture asks for the stronger polynomial divisibility statement.