Walk dimension and vanishing curve modulus in metric measure spaces

On a regular local $p$-Dirichlet space supporting a Poincaré inequality and a capacity upper bound, we characterize the existence of curve families of positive $p$-modulus in terms of the small-scale behaviour of the scaling function. As an application, we show that, on an Ahlfors-regular metric measure space carrying a strongly local regular Dirichlet form with sub-Gaussian heat kernel estimates, the comparison between the walk dimension and $2$ determines whether the given metric attains its Ahlfors-regular conformal dimension.