Existence conjecture for the Gauss image problem under the weak Aleksandrov condition

Let $\mu$ and $\lambda$ be Borel measures on $S^{n-1}$, with $\lambda$ absolutely continuous. Say that $\mu$ is weak Aleksandrov related to $\lambda$ when it satisfies the weak Aleksandrov condition relative to $\lambda$. Existence conjecture. If $\mu$ is not concentrated on a closed hemisphere and is weak Aleksandrov related to $\lambda$, then there exists a solution to the Gauss image problem. This would extend the existence result for the classical Aleksandrov condition to the weak Aleksandrov setting when one measure is absolutely continuous; the source presents this as an interesting question, and no resolution is supplied.