A new Banach space with Valdivia dual unit ball (Q1852718)

As is well known, a Banach space with a Corson weakly* compact dual unit ball admits a projectional resolution of identity (PRI). Answering a question of \textit{M. Fabian, G. Godefroy} and \textit{V. Zizler} [Bull. Pol. Acad. Sci., Math. 47, 221-230 (1999; Zbl 0946.46016)] the author constructs a Banach space (which is \(C[0,\omega_1]\) in some equivalent norm) having no PRI. He also poses a question: Is there a Banach space with a Valdivia weakly compact dual unit ball which admits no PRI in any equivalent norm?