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Answer: This scenario appears absurd to us too.
Unless the State rules in Hawaii state so, it makes no legal sense that a foreign authority would have the jurisdiction and competence to void or in any other way invalidate a legally issued driver’s license from another state or nation.
Just seek a second opinion by asking to see her Supervisor about it. If you meet them and they are also saying the same thing, you could try a different DMV and hopefully get a better and more “reasonable” response. If for some reason you can’t approach another DMV at this time, then allow them to go ahead and then “invalidate” the Hungarian driver’s license. Most likely they will do this by cutting the edges of the license and by stamping “cancelled” across it but do not throw the Hungarian driver’s license away.
Getting a US driver’s license for the state you currently reside is of paramount importance for now for obvious reasons. If at some point in the future you are back in Hungary and need to use a Hungarian driver’s license, just show them the “invalidated” old license and apart from a few laughs that would get and the chance of instant popularity, there should be little or no trouble renewing or reissuing you a new one.