Central binomial sum evaluation involving zeta values and powers of pi

Let

$$S=\sum_{k=0}^{\infty}\frac{\binom{2k}{k}}{2^{2k}(2k+1)^4}.$$

Central binomial sum conjecture.

$$S=\frac{1}{48}\left(6\pi\zeta(3)+4\pi\log^3 2+\pi^3\log 2\right).$$

This identity is obtained by evaluating an arcsine logarithmic integral and was reported as having only computer verification in the source; its status therefore remains open.