BALTIC-WAY 2022 Problem 3

We call a two-variable polynomial $P(x, y)$ secretly one-variable, if there exist polynomials $Q(x)$ and $R(x, y)$ such that $\deg(Q) \ge 2$ and $P(x, y) = Q(R(x, y))$ (e.g. $x^2 + 1$ and $x^2y^2 +1$ are secretly one-variable, but $xy + 1$ is not).

Prove or disprove the following statement: If $P(x, y)$ is a polynomial such that both $P(x, y)$ and $P(x, y) + 1$ can be written as the product of two non-constant polynomials, then $P$ is secretly one-variable.

Note: All polynomials are assumed to have real coefficients.