I think I see the source of the problem. The murata module firmware calls at_Receive asynchronously when it receives a LoRa downlink:
static void LoraRxData(lora_AppData_t *AppData)
set_at_receive(AppData->Port, AppData->Buff, AppData->BuffSize);
To fix this, we need an option that allows us to turn off the call to at_Receive() in the murata module and add a function to the MKRWAN API that allow us to call this function from our application. That way, when we sleep, we will not miss the asynchronous transmission of data from the module.
I opened an issue on Github.