Thanks again for the pictures and the video.
Concerning the keepouts under the connector: 1. At the north boundary I would pull the edge up a little further north away from the northwestern differential pair.
2. At the west boundary I see your point regarding layers 4 and 5. Looks like you have made a good solution. I suppose you could add 5mil additional overlap. How much overlap does it currently have? How much opening from the edge of the keepout on layer 4 to the edge of the closest connector pads?
I would vote to keep the layer 5 holes under the layer 6 ESD pads for the same reason we added them to layer 2 for the ESD pads on layer 1.
Some of the adjustments on layer 6 might be taken care of by modifying the net groups to create an "HDMI High-Frequency" group which contains only the differential pairs {HTX2P, HTX2N, HTX1P, HTX1N, HTX0P, HTX0N, HTXCP, HTXCN}, apply the 15mil conditional clearance rule to that group. Then see what issues remain.