On the weak and pointwise topologies in function spaces

For a compact space K we denote by C_w(K) (C_p(K)) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. During the talk I would like to discuss the following basic question which seems to be open:
Suppose that K is an infinite (metrizable) compact space. Is it true that C_w(K) and C_p(K) are homeomorphic?
I will show that the answer is "no", provided K is an infinite compact metrizable C-space. In particular the proof works for any infinite compact metrizable finite-dimensional space K.