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There is probably a lot of misunderstanding, regarding, showing your LTC, while carrying, when interacting with LEO. My understanding, ( and Charles, please weigh in if I am mistaken) when you are stopped/ detained, for any violation, i.e., a traffic stop, you are required, to show your LTC, but, if it is a situation, described by the OP, it is not required, unless, asked for, by the LEO, in all of my traffic stops, since obtaining, my LTC, I have produced, it, along with my DL. In all cases, it was handed back, with not much more than a cusory, glance. I have absolutely no problem, showing any LEO, my LTC, and whether or not, it is required, is not an issue, for me.
All of my interactions, with LEO, have been positive, but, I have never been asked to produce my LTC, even when OC, by any LEO. JMHO