I spoke with a LATAM representative other the phone and he didn't seem to be aware of that problem. Have been in touch with BA on twitter and here's the answer I had :
Hi Guillaume, I've spoken to our oneworld team, who have checked on this with LATAM. They've advised that they're unsure why your card status isn't showing and are escalating it internally for investigation, however this may take up to 7 days. They've also advised that, if you go to the oneworld desk at JFK tomorrow, they'll be able verify your status. ^BS
I think this is a good thing they have such a dialogue between airlines of a same alliance and I hope they can escalate the problem to the concerned department. Flying out of JFK tomorrow so I will see how the problem is handled by airport agents.