Even if the easy side knows (through STUN) that its public address is 2.2.2.2:1234, that doesn't mean it'll accept any packets addressed to 2.2.2.2:1234.
For it to accept packets from e.g. 5.5.5.5:7890, it first needs to send a packet there, to create the connection table entry in the firewall.
But... it doesn't know what port 5.5.5.5 is going to be sending from (since that's the whole point of 5.5.5.5 having a "hard", i.e., destination-dependent-mapping, NAT), so it can't preemptively send a packet to open the firewall.
At least, this is my understanding.
So, we're left with two options: make one of the NATs vanish via the port-mapping protocols, or use the birthday paradox to find a usable ip:port in reasonable time.
What I got was that "hard" NAT is hard because neither peer is able to obtain the hard NAT's public IP/Port pair: The peer behind the hard NAT can't ask a STUN server because that server would see a different pair while peer behind the easy NAT can't receive any packets.
I think it's interesting that the peer behind the hard NAT could actually produce a packet that would contain all necessary information by sending something to the easy NAT's address - but neither peer could access this packet. You'd have to call the NSA to monitor the packet in-flight...
I wonder if you could somehow fiddle with the max-hop header to have the packet be sent back to the hard NAT's peer though, sort of like traceroute works.
According to the author of dublin-traceroute, it used to be that you could use ICMP TTL Exceeded errors to your advantage, because buggy NATs didn't translate ICMP payload (which contains the IP+UDP header of the packet whose TTL hit zero).
IOW, you could send a probe packet to the easy ip:port, with a TTL low enough that the packet expires somewhere between the two NATs, and listen for the returning ICMP error. Its payload would contain the public ip:port for your session on the hard NAT, which effectively gives you an accidental STUN server that gives you correct answers, even for a hard NAT.
I _believe_ this trick no longer works reliably, because the "NAT Behavioral Requirements" family of RFCs started laying down rules for what well-behaved NATs should do, and afaict, inexplicably, vendors listened. REQ-4 of RFC 5508 says that NAT devices MUST translate the payload of ICMP errors according to the appropriate NAT mapping, to preserve the end device's illusion that no NAT exists. So, these days, the ICMP error you get back contains the unhelpful LAN source ip:port :(
I still want to experiment with that some more, because if it's unusual but not unheard of... Well, it's another technique to grab a little bit more of the long tail :)
That got a good laugh out of me.
That idea sounds really clever, but even if NATs wouldn't mangle the ICMP reply, don't you need elevated privileges to receive it in your application? And I wouldn't be too surprised if a bunch of devices would just drop it entirely.
Another thing that just came to my mind: Do any of those hard NATs assign the outgoing port in a predictable pattern for some inexplicable reason? I'd guess no, since those already seem to be security focused and not randomizing the port sounds stupid, but you never know... Would be a cool last resort for hard/hard.