How to compute $\sum_{n=1}^\infty\frac{H_n^2}{n^32^n}$?

Can we evaluate $\displaystyle\sum_{n=1}^\infty\frac{H_n^2}{n^32^n}$ ?

where $H_n=\sum_{k=1}^n\frac1n$ is the harmonic number.

A related integral is $\displaystyle\int_0^1\frac{\ln^2(1-x)\operatorname{Li}_2\left(\frac x2\right)}{x}dx$.

where $\operatorname{Li}_2(x)=\sum_{n=1}^\infty\frac{x^n}{n^2}$ is the dilogarithmic function.

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