RCD structure of the transverse factor in local almost-splitting

Determine whether the metric-measure factor W arising in the local almost-splitting theorem is an RCD(K,N) space when a pointed RCD(K,N) space is locally close to a product \mathbb{R}^k\times W.

Background

The almost local splitting theorem shows that suitable harmonic maps produce local pointed measured Gromov–Hausdorff closeness to a product of Euclidean space with a metric-measure factor W. Because the theorem is formulated locally, the authors cannot directly apply the global-to-local and local-to-global principles needed to identify W as an RCD space.

The unresolved issue is relevant to understanding whether local quantitative splittings inherit the full synthetic lower-Ricci and upper-dimension structure in the transverse directions. The authors note that when k equals the essential dimension, the factor can instead be taken to be a point.