Equivalence of asymptotic average-case approximability and asymptotic disintegrability

Let $(\mathfrak{X},d,\mu)$ be a metric Polish probability space. Equivalence conjecture. The space $(\mathfrak{X},d,\mu)$ is asymptotically average-case approximable if and only if it is asymptotically disintegrable. Asymptotic disintegrability is already known to be sufficient for asymptotic average-case approximability, while the converse is motivated by a counterexample showing failure of both properties and remains unproved.