It was shown in [BFJ15, §8.1] that uniqueness follows from a result of Yuan and Zhang in [YZ17]. To prove existence of a $\theta$-psh solution $\varphi$, we use the variational method of Boucksom, Favre, and Jonsson. The basic tools needed here are upper semicontinuity of the energy [BFJ15, Prop. 6.2], the compactness theorem [BFJ16a, Thm. 7.10], and the differentiability of $E\circ{P}_{\theta}$. As explained above, all these results are available in our setting. It remains to see that $\varphi$ is continuous and this is done by estimates in the spirit of Kolodziej as in [BFJ15, §8.3]. ∎