On decomposition problem for distribution functions of class 𝑄

We consider a new class Q Q of distribution functions F F that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions F 1 F_1 and F 2 F_2 such that F 1 = F ∗ F 2 F_1=F*F_2 . A distribution function of the class Q Q is quasi-infinitely divisible in the sense that its characteristic function admits the Lévy–Khinchine type representation with a “signed spectral measure”. The class Q Q , being a natural extension of the class I I of infinitely divisible distribution functions, is actively studied now and it finds various applications. In 2018, Lindner, Pan, and Sato [Trans. Amer. Math. Soc. 370 (2018), pp. 8483–8520] formulated the open question: is it true that if F ∈ Q F∈ Q and F = F 1 ∗ F 2 F=F_1*F_2 with some distribution functions F 1 F_1 and F 2 F_2 , then F 1 ∈ Q F_1∈ Q and