As open carry is legal in Connecticut, I cannot say it would be appropriate or reasonable to allow police to demand a permit of a person who is acting within the law.
That being said, open carry is uncommon so, right or wrong, people may be concerned by the sight of a gun.
I believe the only constitutionally acceptable remedy is to require permit holders to present their permit during any interaction with police. A fine difference from police demanding the permit, but this approach is already in use by other states and allows police to examine the permit without running afoul of unreasonable seizure.
Or simply change the current permit to only allow concealed carry. Then any intentional display of a firearm can be investigated as a possible crime.
The bottom line is there are ways to solve the problem that do not run afoul of Fourth Amendment issues. The typical Connecticut approach is to find the most difficult, burdensome, and unreasonable path.
Edward Kelly lives in Branford.