FWIW, something I once saw a "CMS" built for IIS do that horrified me... It had a setting to automatically set the rel canonical URL to self-reference the url of the page it was on, *no matter what it was*.
This same CMS, and I won't reveal what it was, also felt that the main task of optimization was writing the URL to include all of the keywords in the page title... and it also included meta keywords (this was in the past 5 years or so, which made it amazingly out of step). It also included in the meta keywords list the name of the CMS system and its manufacturer, as if that plug for the home team on unrelated sites through the web was going to help the credibility of the company. These particular keywords were built into the system and couldn't easily be removed.
I mention all of the above, which are essentially off-topic, just to emphasize how bad its choices were... including the auto self-referencing.
I'm not, btw, opposing self-referencing canonical urls assuming they can be manually controlled and adjusted. An argument can be made for self-referencing canonicals when properly set, but I'll stay out of that one.
But one can test it out and see if it makes a difference.
That's likely to be a long term and inconsistent test, so long term and inconsistent that I'd say that such a test was meaningless.