If by “proven”, you mean in the sense of a mathematical proof, of course never. But “proving” in the courtroom sense of “beyond a reasonable doubt” is still unlikely, but not so far-fetched.
There is a wealth of forensic evidence making a very strong case that the shroud was wrapped around a man who died as a result of crucifixion (see the many Shroud web sites for details). There is significant, but weaker evidence tying the shroud to Jerusalem in Jesus’ day (pollen, minerals, absence of vanillin, etc.) The C-14 dating that supposedly dated the shroud to the middle ages has now been largely debunked.
As to tying it to Jesus specifically, there are several details (whipping, crown of thorns, legs unbroken) that are described in the Gospels but not usually seen with crucifixion. Beyond that, I resort to Occam’s razor - why would the shroud from another victim have been preserved? When one looks at the totality of evidence, that the Shroud was Jesus’ seems to be the simplest explanation that fits the facts.
How many other fabric items of the same construction still exist from 2000 years ago still exist?
Scientifically, I would not refer to the 1988 C14 test as “debunked” as if it were pseudoscience, but rather as “falsified”. In other words the conclusions have been proved to not be true by other scientific methods. The primary reasons being that the 1988 C14 tests were falsified by showing that the protocols of sample testing were not followed and the correctly done testing actually tested a sample area of the Shroud that had been repaired that included some later introduced cotton rewoven threads of sufficient quantity to skew the testing enough invalidate the test.
As a mater of fact any introduced material of a later than original carbon would skew the results and falsify the results. In this case no less than six different means and peer-reviewed studies falsified the C14 tests.