Khintchine-type conjecture for monotone convolution

Let $\mu,\mu_n\in {\rm \bf P}(\mathbb{R})$, where $n\in\mathbb{N}$, and let $\{k_n\}_{n\geq1}$ be a sequence of strictly increasing natural numbers. Monotone Khintchine conjecture. If

$$\mu_n^{\rhd k_n}\overset{\rm w}{\rightarrow}\mu,$$

then $\mu\in {\rm \bf ID}(\rhd)$.

This is the Khintchine-type assertion for identical factors in monotone convolution. The source explicitly says that the theorem remains open in this setting.