Preservation of complete metrizability under uniformly continuous surjections or uniform homeomorphisms between Dp-spaces

Determine whether, for (separable) metrizable spaces X and Y and a uniformly continuous surjection or a uniform homeomorphism 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

Marciszewski and Pelant posed this problem, which directly impacts whether scatteredness results can be extended from the linear case to the uniformly continuous case, since metrizable scattered spaces are completely metrizable.

A resolution would clarify the behavior of underlying spaces under uniform morphisms of function spaces, an issue central to the broader Cp-theory.