Coincidence of analog and distributional category and topological complexity

The paper defines analog sectional category using the minimal support of a continuous choice of fiberwise probability measure, and denotes the resulting analog versions of category and sequential topological complexity by $\mathrm{acat}(X)$ and $\mathrm{ATC}_r(X)$, respectively. Distributional category and topological complexity are the corresponding notions introduced in the cited contemporaneous work.

Coincidence conjecture. The analog and distributional notions of category and topological complexity coincide on the class of metrizable spaces.

The conjecture reflects the close similarity between the two constructions, although their motivations differ. The source presents it as an expectation and does not provide a resolution.