Density conjecture for QID distributions with finite quasi-Lévy measure

Let $d\in\mathbb{N}$. A probability distribution on $\mathbb{R}^{d}$ is QID if it is quasi-infinitely divisible. A quasi-Lévy measure is finite when it has finite total variation, and the Gaussian variance is zero when the Gaussian component vanishes.

Density conjecture. The class of QID distributions on $\mathbb{R}^{d}$ with finite quasi-Lévy measure and zero Gaussian variance is dense in the space of probability distributions on $\mathbb{R}^{d}$ with respect to weak convergence.

This conjecture concerns approximation of arbitrary probability distributions by quasi-infinitely divisible distributions with restricted Lévy–Khintchine characteristics. The supplied context introduces it but gives no evidence of a resolution, so its status remains open.