I have asked a series of questions concerning capabilities of a certain class of exotic automata which I have called min-heap automata; the original question, and links to others, can be found here.
This question concerns the computational power of type-2 min-heap automata, which were suggested by Raphael as a more natural kind of computing device. The class of languages which can be accepted by such automata is a proper superset of the set of context-free languages; which leads me to my question.
Is $HAL_2$ (the set of languages accepted by nondeterministic type-2 min-heap automata) a proper subset of $CSL$ (the set of context-sensitive languages), or not? Note that it is already known (shown by Raphael in the linked question) that there are $CSL$ languages not in in $HAL_2$.
This is one of the last questions I intend to ask about these automata. If a good answer can be found to these (and other) questions, my curiosity will be completely satisfied. Thanks in advance and for all the hard work so far.