I conjecture a stronger statement: that the function $F(x,y)$ of two variables $x,y$ obtained from $F_n(x)$ by substituting $y$ for $x^{2n}$ is convex for $0<y<x<0$.  While this reduces to checking positive definiteness of 
a matrix of bivariate polynomials of degree about 20, obtained from the Hessian of $F$ by clearing common (positive) denominator, this should be possible to make to work by standard real algebraic geometry tools.
Here is an (ugly) plot of $F$:

[![plot on [0,1]x[0,1]][1]][1] obtained by exporting the plot from jmol in Sage(math) 

    sage: var('x y')
    sage: F(x,y)=log((x+y^2)*(1+y)*(1-x*y)/(x*(1+x*y)*(1-x^2*y)))
    sage: plot3d(F,[0,1],[0,1])

 
  [1]: https://i.sstatic.net/l9j8a.jpg