Here is an interesting question.
Let $G$ be a locally soluble group. We know that if $G$ satisfies $Max$ (the maximal condition on subgroups), then $G$ is trivially soluble. On the other hand, if $G$ satisfies $Max_{n}$, we know that $G$ CAN be insoluble.
Now, we have that $Max<Max_{sn}<Max_{n}$, so how does $Max_{sn}$ act like? Can we say that all locally soluble groups with $Max_{sn}$ are soluble or does it exist one of them which is not?
[This question is in the area of looking for similar behavior of different properties in similar context. In this case, for example, $Max=Max_{n}$ for locally nilpotent groups]
Definitions
Locally soluble: Every finitely generated subgroup is soluble
$Max_{sn}$: Every chain of subnormal subgroups of $G$ is finite
$Max_{n}$: Every chain of normal subgroups of $G$ is finite