Does uniform inverse boundedness imply c-goodness?

Determine whether every uniformly continuous inversely bounded mapping T : Dp(X) -> Dp(Y), where Dp(X) denotes either Cp(X) or C*(X) endowed with the topology of pointwise convergence, is c-good for some constant c > 0.

Background

The authors show that if T : Dp(X) -> Dp(Y) is a uniformly continuous inversely bounded surjection between function spaces over metrizable spaces, then several dimensional-like properties (including zero-dimensionality and strong countable-dimensionality) transfer from X to Y. Earlier results established such transfers under the stronger c-good condition.

It remains unknown whether inverse boundedness alone implies c-goodness. Resolving this would unify the conditions under which these preservation results hold and clarify the relationship between these two map regularity notions.