Series with summands involving higher order harmonic numbers

For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\sum_{0<k\le n}\frac1{k^m}\ \ (n=0,1,2,\ldots).$$ We discover exact values of some series involving harmonic numbers of order two or three. For example, we conjecture the curious identity $$\sum_{k=1}^{\infty}\frac{(3k-1)(-8)^k}{k^3\binom{2k}k^3}\left(H_{2k-1}^{(2)}-\frac54H_{k-1}^{(2)}\right) =-2\beta(4),$$ which provides a fast converging series for computing $\beta(4)=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^4}$.