This has been one of those 2A wet dreams for 30 years now. Its not going to happen because it creates an internal conflict from a conservative government point of view (rather than from any other kind of "conservative" point of view - and people make the mistake of conflating or those different aspects of "conservative").
On the one hand, we can all point to things like driver's licenses, marriage licenses, etc and claim concealed carry permits should be the same way.
On the other hand we have state's rights. There is for example (non commercial) drivers license reciprocity only because of mutual agreements among states, usually informal. States rights is the conservative government argument that historically has prevailed on issues like this that are not viewed as specifically violating a constitutional right.
Issuance of marriage licenses is also a right granted to the state and as you well know it can be controversial in some situations with some states not having the same standards as others and occasionally resulting in federal court cases to determine whether it crosses the lines of violating a constitutional right.
The reality is that a much stronger case can be made for driver's license reciprocity due to interstate commerce interests, and marriage license reciprocity due to the chaos that would result otherwise, than can be made for concealed carry reciprocity.
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At best we might get clarification in the form of a federal all defining what you can carry in another reciprocal state. However, that's still a huge stretch and falls in the "you wont like what you get" category. For example if carrying a hollow points in a 17 round magazine is legal under your permit in your home state, a federal law could state it would also be legal in a state that bans hollow points and magazines with more than 10 round capacity in a state with a reciprocity agreement. Similarly, similar logic could be applied in where its legal to carry in a state rather than having a hodge podge of different state rules.
The argument there is that states can and should be able to determine what their residents can legally carry and where they carry it as a state right, but they should not be able to dictate what the residents of other states can carry and where. Good luck with that.
The problem of course is if a federal law like that came to pass, it would be one more factor states would then add to the burden of establishing reciprocity agreements in the first place, and we'd see a lot fewer reciprocity agreements.
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Constitutional Carry also isn't a solution. As implemented in a number of states already, states vary on whether it applies to everyone or just their state's residents.
There's also the practical matter that persons with concealed carry permits have been found to be 6-7 times less likely to commit any felony or misdemeanor than law enforcement officers. Most states require at least some level of training in firearms safety and or the laws pertaining to the use of deadly force. Constitutional Carry laws by definition let people carry with no guarantee they understand safe gun handling or what they can and cannot due when it comes to the use of deadly force. In the long term there will inevitably be more "bad" shoots, that have the potential to swing the pendulum way back in the other direction.
And of course, carrying under constitutional carry doesn't give you any state reciprocity, unless that state recognizes constitutional carry for non residents. Consequently some states with Constitutional carry still offer concealed carry permits. SD for example offers three levels of concealed carry permits with differences in if and how many states honor the permit.