In Wifi WPA, I understand that during association, a 802.11 client and AP negotiate a Pairwise Transient Key (PTK), using which the Groupwise Transient Key(GTK) is provided to the station.

I understood recently that all communication from/to this client, including broadcast from the client to all other stations happen through the AP.

Why then does the client need to encrypt the broadcast using the GTK? Why can't the client just encrypt the broadcast frame using its PTK, and the AP decrypt it, then encrypt it using each of the other clients' PTK before sending it out to them?