Conjecture on logarithmic integrals with powers of \log$1-x$ and \log$1+x$

Let $n\geq 1$. Consider the logarithmic integral

$$\int_0^1 \frac{\log^{n+1}(1-x)\log^n(1+x)}{1+x}\,dx.$$

Logarithmic-integral algebraicity conjecture. For every $n\geq 1$, this value belongs to the algebra over $\mathbb{Q}$ generated by $\pi^2$, $\log 2$, and $\zeta(n)$ for odd $n$. The cases $n=2,3,4,5$ are reported as verified or explicitly evaluated, but a unified proof for general $n$ remains elusive.