Arduino Nano no serial communication SIM800C

I am trying to get my SIM800C to talk with my Arduino. There is no communication happening, though.

#include <SoftwareSerial.h>

SoftwareSerial at(2, 3);

void setup() {
  Serial.begin(9600);
  at.begin(9600);
}

void loop() {
  // try every 2 seconds
  delay(2000);

  Serial.println("sending AT ... ");
  at.println("AT");

  while (at.available() > 0) {
    Serial.write(at.read());
  }
}

I am not able to get an OK back. SIM800C is supposed to detect the baud rate by itself.

I am sure there has to be a simple stupid mistake. I just don't know what to do at this point. I obviously already checked for cable break. Out of desperation I already tried to switch RX and TX. I also tried different baud rates (whatever is within the usual limitations of SoftwareSerial) but it should automatically detect it once a couple of AT commands got in anyway.

Weird enough, the pin PWX on the SIM800C needs to be hooked up to a GND to work. It started blinking every second now and is responding to AT commands.

Also it turned out that this specific module does not ship with autobauding enabled, as stated by the SIM800C documentation. The correct baud rate is 115200.