Attainment and (sub)differentiability of the supremal convolution of a function and square of the norm

Let X be a Banach space whose norm is LUR. Under some mild assumptions, it is shown that the supremal convolution of a fairly general function on X and the square of the norm is generically strongly attained. This contains a recent result of Bandyopadhyay and Dutta on farthest distance functions.