I've been working on a project and proved a few relevant results, but got stuck on one tricky problem:

>**Conjecture.** If $2\leq n\in\mathbb{N}$ and $0<x<1$ is a real number, then 
$$F_n(x)=\log\left(\frac{(1+x^{4n-1})(1+x^{2n})(1-x^{2n+1})}{(1+x^{2n+1})(1-x^{2n+2})}\right)$$
is a convex function of $x$.

>Can you help with a rigorous proof or valuable tools? I have a heuristic argument.

**Further motivation.** If you succeed with this, then I'll be honored to have you as a [co-author in this work][1]. The problem itself can be found in Section 4.

**Note.** Write $F=\log\frac{P}Q$, then $F''>0$ amounts to the positivity of the polynomial
$$V:=PQ^2P''+(PQ')^2-P^2QQ''-(P'Q)^2.$$


[1]: https://www.math.temple.edu/~tewodros/Catalan_Convexity.pdf