Equality of average-order constants for lucky omega functions

The lucky divisibility functions are defined by

$$\overset{*}{\omega}(n):=\sum_{l\overset{*}{|}n}1,\qquad \overset{*}{\Omega}(n):=\sum_{l\overset{*}{|}n}\overset{*}{\nu}_l(n),$$

where $\overset{*}{\omega}(n)$ counts distinct lucky divisors and $\overset{*}{\Omega}(n)$ counts them with their lucky order. Lucky omega constants conjecture. There exist constants $A,B\in\mathbb{R}$ such that

$$\sum_{n\leq x}\overset{*}{\omega}(n)\sim Ax\log\log x,$$ $$\sum_{n\leq x}\overset{*}{\Omega}(n)\sim Bx\log\log x,$$

and, if these asymptotics hold, $A=B$. This asks whether the two lucky analogues of the classical prime omega functions have the same average-order constant; the source presents it as a question and gives no resolution.