fob stopped working after battery replacement

Hi

I replaced batteries on both intelligence key fobs. due to both reported low battery.
one of the fobs doesnt lock or unlock the car anymore however if i put the fob close to the start button the engine will turn on
I tried replacing the battery again with a second new one still same issue.

I took the car and keys to the dealership they said they tried reprogramming it but they noticed the antenna on the key is faulty.
I took both keys apart and i dont see any missing or any physical damage to any components. i did a simple continuity&resistance on the bigger components and compared working and nonworking fobs and they are identical. no components are loose either when i push on them.

any ideas which area to look at if the dealership are telling the truth that the "antenna" is faulty.