C-space × locally compact space: product invariance

Determine whether, for any C-space E and any locally compact space L, the product E × L is a C-space; and similarly, determine whether, for any Cs-space E and any locally compact space L, the product E × L is a Cs-space.

Background

Locally compact spaces are known to be C-spaces. The paper establishes several product non-invariance results for related classes, but leaves open the specific case of taking the product of a C-space (or Cs-space) with a locally compact space.

Clarifying this case would illuminate whether local compactness of one factor can salvage product invariance for closed graph spaces, which is relevant to understanding the limits of extending Banach-type arguments in general topology.