Is there a computing problem which is in quasi-polynomial time but is (maybe) not in $\beta P$?
While I don't know a specific (conjectured) example in $QP-\beta P$, there is still rather compelling evidence that $\beta P$ is a proper subset of $QP$. Namely, these classes behave very differently in their relationship to $NP$:

$\bullet$ It is obvious from the definition that $\beta P\subseteq NP$.

$\bullet$ On the other hand, $QP\subseteq NP$ is not known, and it would be very hard to prove, since it implies $P\neq NP$. (In fact, it is an even stronger statement than $P\neq NP$.)

Such a very different behavior relative to $NP$ seems to provide a fairly strong reason to believe that $\beta P\neq QP$.