Dual Kadec-Klee property and fixed points

A Banach space X is said to have the dual Kadec-Klee property iff on the unit sphere of the dual space X⁎ the weak*-topology coincides with the norm-topology. A. Amini-Harandi and M. Fakhar extended a theorem of B. Ricceri concerning the Kadec-Klee property by showing that if X has the dual Kadec-Klee property then for every compact mapping φ:BX⁎→X∖{0} there exists some f in the unit sphere of X⁎ such that 〈f,φ(f)〉=‖φ(f)‖ and conjectured that this yields a necessary and sufficient condition for X to have the dual Kadec-Klee property. We prove here that it is almost the case: we introduce a weakening of the dual Kadec-Klee property, here called the NAKK* property, and show that this property is equivalent to the above property about compact mappings from BX⁎ to X∖{0}.