On the weak${}^{\ast }$ separability of the space of Lipschitz functions

We conjecture that whenever $M$ is a metric space of density at most continuum, then the space of Lipschitz functions is $w^{\ast }$-separable. We prove the conjecture for several classes of metric spaces including all the Banach spaces with a projectional skeleton, Banach spaces with a $w^{\ast }$-separable dual unit ball and locally separable complete metric spaces.