Can TC improve the IQ of original lens?

Back to the original question.

There are optical precedents. In microscopy a tube lens was dsigned to correct uncorrectable errors in the microscope objective. In astronomy teleconverters are also used to perform field flattening to work better with CCD chips. Both of these improvements are theoretically possible with a matched converter but the chances of finding a random match are pretty slim. Contrast and optical alignment will always be marginally worse. But we just need to try, just in case