The Fibonacci polynomials are defined recursively by $F_0(x)=1, F_1(x)=1$ 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? $$\int_0^1F_n(k+nz)\,dz$$