Question: Given a non-deterministic timed automata with 1 clock(1-NTA) and reachability acceptance condition, decide whether there exists an equivalent history deterministic timed automata with k clocks ($k$-HDTA), where $k$ can be fixed or arbitrary?
Motivation: The classes of 1-NTA, deterministic timed automata (DTA), history-deterministic timed automata (HDTA) all have decidable language inclusion problems, though the complexity of inclusion for 1-NTA is non-primitive recursive. However, the class 1-NTA is incomparable to DTA and HDTA.
Whether a given 1-NTA is equivalent to some $k$-DTA is decidable, but it the problem becomes undecidable if the number of clocks in the DTA is not fixed beforehand. The proposed question is simply extending this to HD case instead of deterministic.
Remarks: It is unknown whether HDTA is a strictly bigger class than DTA. Showing that the two classes (DTA and HDTA) are expressively equivalent would settle the problem by reduction to the DTA case.