This is pretty clear to me. If you wanted to make it sound like you're claiming, you'd phrase it something like "Texas may well execute soon an innocent man, the first in US history, on the false premise of "shaken baby syndrome.""
As it is, the first sentence very specifically scopes down this case to "shaken baby syndrome."
The disproven ‘shaken baby syndrome’ may for the first time result in a man’s execution in Texas. His innocence …
The odd wording is specifically there to mislead someone.
It is clear that it is meant the shaken baby syndrome is what is the first here, but still it could be taken the other way, and it ends up downplaying how contemptible the State's actions are, even though the poster really wants to highlight that.
I guess I would prefer something like -
"Texas, continuing its tradition of executing innocent people, will for the first time use the dubious science of Shaken Baby Syndrome as its cause."
It's not just the post, everybody does it whenever they say "Texas might execute an innocent person" they ignore that Texas has executed people that were extremely likely to be innocent (so likely that it is reasonable just to describe them that way - in case anyone has problems with my phrasing here)
What is odd about the original wording? If anything, the one you present feels more formal/awkward compared to normal conversational English.
Personally I think it's absolutely ludicrous to believe that anyone would be trying to suggest that only now, in the year 2024, has a possibly-innocent person been executed in either Texas or the US. Like, you'd need crazy strong evidence to make that claim, while the post in question seems very specific to shaken baby syndrome.
It feels like an intentionally un-charitable, adversarial reading.
This may be a regional thing but the second just sounds better to me:
Texas may well execute soon an innocent man
Texas may soon execute an innocent man
“on the false premise” is also strange. I don’t think it’s really a logical argument so much emotional manipulation around an images of a dead and bruised body that was compelling. Kids are tough, the idea of shaking hard enough to damage a neck is plausible but to cause serious bruises is difficult to believe. Add in evidence of a serious medical condition and you get a different picture than just a false premise.> It feels like an intentionally un-charitable, adversarial reading.
Saying this might be the first isn’t strictly speaking wrong as far as I can tell. I don’t know of anyone proven to be innocent that’s been executed. Sure, it’s likely happened several times, but I can’t exactly put a firm number so it may well be zero.