I've been using Club3D DP to HDMI adapters for monitor hook ups. So far, so good.
They don't have to be "active", your card will take care of that.
But pay attention to the version numbers: DisplayPort 1.2 to HDMI 2.0 at the least. I think you can now get DP 1.4 -> HDMI 2.0a.
Curious, what model is the Sony? From its viewpoint, the DP on the card should be just like any other HDMI source.