The Relatively Prime Conjecture for factorial polynomials

For ${\bf a},{\bf b}\in\mathbb N^2$ with ${\bf a}\ne{\bf b}$, write ${\bf c}={\bf a}-{\bf b}$ and define, for $n\in\mathbb N$,

$$P_{{\bf a},{\bf b},n}(X)=\sum_{k=0}^n\frac{(b_1n+c_1k)!(b_2n+c_2k)!}{k!(n-k)!}X^k.$$

Relatively Prime Conjecture. For all such ${\bf a},{\bf b}$ and all $n\in\mathbb N$, the polynomials $P_{{\bf a},{\bf b},n}(X)$ and $P_{{\bf a},{\bf b},n+1}(X)$ have no common zero in ${\mathbb C}$. By the theorem preceding it, this is equivalent to the Strong Factorial Conjecture for polynomials in two variables consisting of two monomials. The paper proves several special cases, leaving the general assertion open.