Characterization of measure sequences defining a density

Let $G$ be a discrete group, and let $(\mu_n)$ be a sequence of measures on $G$. A sequence of measures defines a density in the sense that the associated upper density is mensural if, for every $t\in G$, it satisfies

$$\lim_{n\to\infty}\sum_{x\in G}|\mu_n(x)-\mu_n(x+t)|=0.$$

Measure-sequence characterization. The sequences of measures defining a density in this sense are precisely those satisfying

$$\lim_{n\to\infty}\sum_{x\in G}|\mu_n(x)-\mu_n(x+t)|=0$$

for all $t\in G$. The question concerns which sequences of measures yield densities, extending the measure-based descriptions of asymptotic and logarithmic densities; the supplied text does not state whether this criterion has been proved or remains conjectural.