I declared my luggage lost by AF on March 25th, I just got a compensation paid yesterday, so 6 weeks later.
Indeed, you file a claim then you just wait.

You can send them messages through your claim but they never answer. Got an email once saying that they regret that I am so impatient and asking not to contact them again until further notice

Finally, one day, you just find out that the case is closed with an amount and that's it.