Rotund Gâteaux smooth norms which are not locally uniformly rotund

In a recent monograph on renorming techniques in Banach spaces, A.J. Guirao, V. Montesinos, and V. Zizler posed the following problem; does every infinite-dimensional separable Banach space admit a norm that is rotund and Gâteaux smooth but not locally uniformly rotund?
The aim of the talk is to provide a positive answer to the above question. All the presented results have been obtained in collaboration with C.A. De Bernardi.