Closed form for the skew-harmonic sum $\sum_{n = 1}^\infty \frac{H_n \overline{H}_n}{n^2}$
Using Cornel's strategy here https://math.stackexchange.com/q/3459740, we have
$$\sum_{n=1}^{\infty} \frac{H_n \overline{H}_n}{n^2}=1+\sum_{n=1}^{\infty} \frac{H_{2n} \overline{H}_{2n}}{(2n)^2}+\sum_{n=1}^{\infty} \frac{H_{2n+1} \overline{H}_{2n+1}}{(2n+1)^2}$$ $$=1+\sum_{n=1}^{\infty}\frac{H_{2 n}^2}{(2n)^2}+\sum_{n=1}^{\infty}\frac{H_{2 n+1}^2}{(2 n+1)^2}-\sum_{n=1}^{\infty}\frac{H_n}{(2 n+1)^3}-\sum_{n=1}^{\infty}\frac{H_n H_{2 n}}{(2 n)^2}-\sum_{n=1}^{\infty}\frac{H_n H_{2 n}}{(2 n+1)^2}$$ $$=\frac{43}{16}\zeta(4)+\frac{3}{4}\log^2(2)\zeta(2)-\frac{1}{8}\log^4(2)-3\operatorname{Li}_4\left(\frac{1}{2}\right),$$
where the hardest sums (the last two ones) use the strategy in this paper On the calculation of two essential harmonic series with a weight 5 structure, involving harmonic numbers of the type $H_{2n}$ by C.I. Valean.
A first note: The hardest sums in closed-form are $$i) \ \sum_{n=1}^{\infty}\frac{H_n H_{2 n}}{(2 n)^2}=\frac{13}{32}\zeta(4)+\frac{7}{8}\log(2)\zeta(3)-\frac{1}{4}\log^2(2)\zeta(2)+\frac{1}{24}\log^4(2)+\operatorname{Li}_4\left(\frac{1}{2}\right);$$ $$ii) \ \sum_{n=1}^{\infty}\frac{H_n H_{2 n}}{(2 n+1)^2}=\frac{1}{12}\log^4(2)-\frac{1}{2}\log^2(2)\zeta(2)+\frac{7}{8}\log(2)\zeta(3)-\frac{1}{4}\zeta(4)+2\operatorname{Li}_4\left(\frac{1}{2}\right),$$ and are immediately extracted by following the strategy in the paper mentioned above. These results I took directly form Cornel's page here and here, to avoid boring calculations.
A second note: From the strategy in this answer here one may obtain immediately that $$\sum_{n=1}^{\infty}\frac{H_n}{(2 n+1)^3}=\frac{45}{32}\zeta(4)-\frac{7}{4}\log(2)\zeta(3).$$
Further, the first two sums from the second line may be immediately reduced to sums that are pretty known and they may also be found in the book, (Almost) Impossible Integrals, Sums, and Series.
A third note: The weight $5$ version of the present series may be found and calculated in the paper Two advanced harmonic series of weight 5 involving skew-harmonic numbers by C.I. Valean. As seen in the paper, the present series can also be approached in a different way.