On my boarding pass this past weekend, on the outbound flights, I didn't get Sky Priority identification on my pre printed boarding pass. At the issuance of my e-boarding pass, no flight segment of four had been upgraded. I did get a gate upgrade on my connecting flight on outbound flight.
On the return, I had already obtained an upgrade on the connecting leg, but no dice on the gate upgrade for the 1st flight. Both e-boarding pass indicated "Sky Priority".
I don't yet know the qualifying event which qualifies you to Sky Priority but I think you get Sky Priority if you get upgraded on at least one of the flight segment. I don't know if flights on inbound flight matter.
Jiburi