Preservation of complete metrizability under inversely bounded uniformly continuous surjections between Dp-spaces

Determine whether, for (separable) metrizable spaces X and Y and an inversely bounded uniformly continuous surjection T : Dp(X) -> Dp(Y), complete metrizability of X implies complete metrizability of Y, where Dp(X) denotes either Cp(X) or C*(X) endowed with the topology of pointwise convergence.

Background

This variant of the Marciszewski–Pelant problem asks about the stronger hypothesis of inverse boundedness. The paper establishes several preservation results under inverse boundedness (e.g., Theorem 1.3 and Theorem 4.6), but whether complete metrizability transfers remains open.

An affirmative answer would strengthen the understanding of how metric completeness interacts with uniform continuity and surjectivity in the Cp/C* framework.