Relationship between SharpP-dimension and SpanP-dimension

Determine whether there is any general implication or separation between SharpP-dimension and SpanP-dimension; specifically, establish whether dim_{#P}(X) ≤ dim_{SpanP}(X) (or the reverse) holds for all classes X, or produce counterexamples.

Background

The paper defines counting dimensions via #P- and SpanP-martingales, generalizing time-bounded dimension. While bounds relative to P- and PSPACE-dimensions are proved, the mutual comparison between the #P- and SpanP-dimensions is unresolved.

Clarifying this relationship would position counting dimensions within the larger landscape of fractal dimensions in complexity.