The radial-limit identity for the universal mock theta function $g_3$ at cubic roots of unity

Let $x,q$ be roots of unity with $(x,q/x;q)_\infty=0$, where $q$ is a primitive root of unity of order $3k$ and $x^{3k}$ is not a primitive sixth-order root of unity.

Radial-limit identity. The following identity should hold:

$$\frac{1}{1-x^{3k}+x^{6k}}\sum_{j=1}^k (-1)^j x^{3j-2} q^{-\frac{(3j+1)j}{2}} \left(q(1+x^{3k}q^k)+x(1+x^{3k}q^{2k})\right) =-x^{-1}+x^{-2}\sum_{j=1}^{3k}(q/x;q)_{j-1}(x;q)_j q^j.$$

This identity was motivated by numerical computations in an overlap case for radial limits of $g_3$; the source presents it as a conjecture, and no proof or resolution is supplied here.