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There is notion of template parameters in the API (CsmTemplateParameter interface, CsmTemplate.getTemplateParameters()). But this API isn't implemented.
In the case project isn't yet completely parsed, code completion shows incomplete results, it doesn't wait for the project to be parsed. The only thing it waits prior then showing its list is parsing of the *current* file. There might be situations when it can wait longer than it's should for the current file, that's because of the parser queue prioritizing issues (see issue #86760) But I can hardly believe it can last a minute. I think most likely the reported has an issue with memory and project size.
Please disregard my last comment. It was for another issue. Sorry.
I see an implementation.