Prevarieties or varieties extending locally pseudoconvex spaces

Determine whether there exists a prevariety or variety of topological vector spaces containing all locally pseudoconvex spaces for which the hemicompactness criterion for the associated free space L_{\mathfrak B}(X) differs from the criteria established for topologies between L_p(X) and L^{(0)}(X) and for V(X).

Background

For a prevariety \mathfrak B of topological vector spaces, the paper considers the free object L_{\mathfrak B}(X). It explains that prevarieties of locally convex spaces yield the same hemicompactness condition as the spaces between L_p(X) and L(X): X must be countable and all compact subsets of X must be finite.

The question asks whether enlarging the class to one containing every locally pseudoconvex space can produce a free space with a different hemicompactness classification. The authors identify the variety generated by locally pseudoconvex spaces and L_0[0,1] as a candidate and discuss its associated free spaces, but do not resolve the general existence question.