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. The problem itself can be found in Section 4.

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