Weak closedness of monotonically infinitely divisible distributions

Let ${\rm \bf ID}(\rhd)$ denote the class of monotonically infinitely divisible probability measures on $\mathbb{R}$, and equip the space of probability measures with the weak topology. Weak-closedness conjecture. The set ${\rm \bf ID}(\rhd)$ is weakly closed.

The class ${\rm \bf ID}(\rhd)$ is a subclass of the measures with univalent Cauchy transforms. Weak closedness would give a useful structural property of monotone infinite divisibility; the source does not state whether this conjecture has been resolved.