That is simply not the case, both ends must be set the same. This isn't an argument worth having if you do not understand this basic networking principle.
Surely you've plugged a device advertising 10/100 into a switch advertising 10/100/1000 and vice-versa and observed that they link at 100M. No?
I will say that leaving a 10/100/1000 device set on autoneg and the other fixed at 100-FD will typically result in a duplex conflict. But that's not what I'm suggesting to be done. Side note, I do believe autoneg and 100-HD typically results in successful link for well behaving clients. Correct? If so that would be yet option as well.
My cheap Linksys LGS308 supports the ability to keep auto negotiate turned and select the advertised speeds (presumably by setting the technology ability bits described in the 802.3 28/40 FLP frames) and effectively making it appear like a 10/100 or 100-only auto device. Unfortunately I don't think the more popular Netgear GS108E has this ability. But I think the GS108T does. If the OP does not have a switch with this capability or willing to purchase one that does then he'll have to consider other options or move on. Though personally I don't see what the big deal is. It's not like the PS4 is going to saturate a 1 gbps connection with any regularity.
BTW...please accept my sincere apology if anything I've posted comes across as argumentative. That's never my desire on this forum or the many others I participate in. My main motivation for posting at all is to learn. So I definitely encourage people to step forward and let me know if I've gotten something wrong. I'd rather take one on the chin and learn something than to continually misspeak.