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Hello, my helpful friends! I have returned to let y'all know that my Shareplay is up and running and working perfectly at home! At my wit's end, my next move was to get my own modem and router. All this time I was under the impression that I had to use the modem/router I was issued by Cox.
Last night I ordered an ASUS CM-16 with an ASUS AC1300 wireless router, and in a few minutes, had it up and running. Tested Shareplay and BOOM. Good to go.
The device I was using (that is now the suspect in the busted Shareplay as well as MMS texts taking forever to go through) is an Arris TG2472, which I will be happily returning to Cox.
I have the same problem as you and my shareplay refuses to work, I even bought the same modem and router you did and still nothing. My speeds are faster but shareplay still refuses to work. My friend was able to connect once and then never again after that. This is really frustrating.
If I still couldn't get it going after buying the new devices, my next move was going to be to replace all Ethernet cables involved. I'm sure I've closed a door on them once or twice. Or more. And sometimes they go bad for no reason. Even brand new ones. It couldn't hurt to try. Also, if you haven't yet contacted Sony tech support, I would do that as well to be absolutely sure your PS4 settings are all correct. Good luck!