Extract imaginary part of $\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$ in closed form

Question

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g.

$\mathrm{Re}[\text{Li}_2(i)]=-\frac{\pi^2}{48}$

Is there a closed form (free of polylogs and imaginary numbers) for the imaginary part of

$\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$

Answer 1

Inspired by this answer and by the comments below it, we could express it in terms of a generalized hypergeometric function as the following:

$$ \Im\left[\text{Li}_3\left(\frac{2}{3}-\frac{2\sqrt{2}}{3}i\right)\right] = \frac{1}{3}\arcsin^3\left(\frac{\sqrt3}{3}\right) - \frac{2\sqrt3}{3}{_4F_3}\!\left(\begin{array}c \tfrac12,\tfrac12,\tfrac12,\tfrac12\\\tfrac32, \tfrac32,\tfrac32\end{array}\middle|\,\frac13\right). $$