real analysis - Infinite Series $sum_{k=1}^inftyleft(zeta(2)-sum_{n=1}^kfrac1{n^2}right)^2$

Evaluate:

$$\sum_{k=1}^\infty\left(\zeta(2)-\sum_{n=1}^k\frac1{n^2}\right)^2$$

Recognizing that $\zeta(2)-\sum_{n=1}^k\frac1{n^2}$ can be written as $\psi_1(1+k)$ where $\psi_1(z)$ is the trigamma function,
What remains to be done is to evaluate:
$$\sum_{k=1}^\infty\psi_1^2(k+1)$$
Mathematica could not evaluate it in a closed form but the source assures that it exists.

If you liked this problem check out Hard Definite integral involving the Zeta function.