The present quest emanates from this study by R. Stanley, including his recent MO question. Define the product (polynomials after full expansion) $$I_n(x)=\prod_{i=1}^n(1+x^{F_{i+1}})$$ based on the Fibonacci numbers $F_n$ with $F_1=F_2=1$. My focus is on the sequence $c_p(n)$, which enumerates the number of monomials, in $I_n(x)$, having non-zero coefficients modulo a prime $p$.
For example, if $p=2$ then $c_2(n)$ starts with $2, 4, 6, 8, 12, 20, 32, 48, 72, 112, \dots$. One may readily translate this problem to counting monomials in the product $\prod_{i=1}^n(1-x^{F_{i+1}})$ (see OEIS A104767) whose coefficients are $-1, 0, 1$ (see this paper).
QUESTION. Given a prime $p$, is it true that the generating function $\sum_{n\geq1}c_p(n)y^n$ is a rational function?
Examples. We have \begin{align} \sum_{n\geq1}c_2(n)y^n &=\frac{2y+2y^3}{1-2y+2y^2-2y^3}, \\ \sum_{n\geq1}c_3(n)y^n &=\frac{2y+3y^3+4y^5}{1-2y+2y^2-3y^3+4y^4-4y^5}, \\ \sum_{n\geq1}c_5(n)y^n &=\frac{2y+y^3+3y^5+4y^7}{1-2y+y^2-y^3+2y^4-3y^5+4y^6-4y^7}. \end{align}
