Nyrath is pointing to ScienceAlert updating "A Physicist Came Up With Math That Shows 'Paradox-Free' Time Travel Is Plausible". It's been two years but nobody has broken the Costa-Tobar logic yet. Google Scholar is pointing to twenty followups, none of which look like debunkers.
I don't think a time-machine can happen. I do wonder about FTL, though. One implies the other; causality is an obvious problem with time-machines, perhaps a non-obvious problem with FTL but still a problem whether or not you understand it (disclosure: I don't).
If causality has workarounds with explicit time-machines, then those same workarounds open a loophole for the implicit time-machine which is FTL.
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