Conjecture on logarithmic integrals in \mathcal{I}_{2n+2}

Let $n\geq 0$, and let $\mathcal{I}_{2n+2}$ denote the space used in the paper for the relevant weight-$2n+2$ logarithmic integrals. Consider

$$\int_0^1 \frac{\log(1-x)\log^{2n}x\log^2(1+x)}{x}\,dx.$$

Logarithmic-integral space conjecture. For every $n\geq 0$,

$$\int_0^1 \frac{\log(1-x)\log^{2n}x\log^2(1+x)}{x}\,dx\in\mathcal{I}_{2n+2}.$$

This is a structural conjecture about the spaces of logarithmic integrals introduced in the paper; the supplied context does not state a proof or resolution.