You create a custom connector outside of a solution in your development environment. The custom connector includes custom code. You plan to deploy the custom connector to another environment by first

Deploying the solution as unmanaged preserves all component data including the custom code files associated with the connector, because unmanaged solutions transport the full connector definition without restrictions. When a custom connector is created outside a solution context and then added to one, the unmanaged deployment method is required to carry the code payload intact to the destination environment.

Managed solutions restrict component modification and do not reliably carry custom code files that were authored outside the original solution context, which is the root cause of the failure.

Publishing the connector in the development environment makes it available for use locally but has no effect on how its custom code is packaged or exported within a solution.

Importing a new solution still does not resolve the issue because the problem is the deployment type, not which specific solution file is used.