Conjectural evaluation of the cubic complementary elliptic-integral moment

Let $K(x)$ be the complete elliptic integral of the first kind, and let $K'(x)$ denote its complementary counterpart. The integral is taken over $0\leq x\leq 1$. Cubic elliptic-integral moment conjecture.

$$\int_0^1 K'(x)^3\,\mathrm{d}x=2K\left(\frac{1}{\sqrt{2}}\right)^4=\frac{\Gamma(1/4)^8}{128\pi^2}.$$

The paper reports this evaluation as numerically verified to extremely high precision, but does not provide a proof; it is presented as the missing relation between two groups of cubic elliptic-integral moments.