The Fibonacci polynomials are defined recursively by $F_0(x)=0, F_1(x)=1$ and $F_n(x)=xF_{n-1}(x)+F_{n-2}(x)$, for $n\geq2$.

While computing certain integrals, I observe the following (numerically) which prompted me to ask:

Question. For $n, k\in\mathbb{N}$, are these always integers? $$\int_0^1F_n(k+nz)\,dz$$

To help clarify, here is a list of the first few polynomials: $$F_2(x)=x, \qquad F_3(x)=x^2+1, \qquad F_4(x)=x^3+2x.$$