Pointwise-versus-weak homeomorphism for general Tychonoff spaces

Determine whether there exist an infinite Tychonoff space X and an infinite-dimensional Banach space E such that the pointwise function space C_p(X) is homeomorphic to the weak-topology Banach space E_w.

Background

The paper proves that if K is compact Hausdorff and E is an infinite-dimensional real Banach space, then no continuous bijection from C_p(K) onto E_w can have an inverse that is continuous at the image of zero. In particular, C_p(K) is not homeomorphic to C_w(L) for any infinite compact Hausdorff spaces K and L.

The authors explain that their proof relies essentially on compactness: compactness makes the relevant finite-subset hyperspaces compact and supplies the accumulation-point and separation arguments used in the pointwise-null cutoff lemma. These mechanisms are unavailable for arbitrary Tychonoff spaces, so the paper leaves unresolved whether the non-homeomorphism result extends from compact Hausdorff domains to all infinite Tychonoff spaces.