Until I get an opportunity to review the design files it is difficult to make detailed recommendations. Have you identified which signals are affected? Have you been able to determine which signals are leaking into the affected signals?
If the image posted with the update[*] is representative of the board geometry, it looks like there could be direct electromagnetic coupling of the 4 HDMI differential pair transmission lines as follows: HTX0 <-> HTX1, HTX2 <-> HTX1, HTX2 <-> HTXC. This could be caused by (and ameliorated by) the following: 1. HTX0 <-> HTX1 Cause: Microstrips are antennae with cross-section consisting of the area between the pair of differential conductors. These two are traveling in parallel on different layers through the ESD section with overlapping antenna area. Possible Remediation: move U10 "up" or "down" far enough relative to the pair on the other side of the pcb that these two pairs don't exhibit any overlapping antenna area 2. HTX2 <-> HTX1 Cause: HTX1P has length correction loop just before reaching the land on CON3 pin 6. I applaud the strategy of clustering impedance discontinuities, thus putting the loop near CON3. The problem is that this expanded antenna cross-section on HTX1 face an HTX2 microstrip antenna. Possible Remediation: Move HTX1P length-compensating loop so it doesn't couple to any other high-speed antenna. 3. HTX2 <-> HTXC Cause: Near edge of vignette, microstrip antennae directly facing each other as they leave the field of view. Possible Remediation: Move microstrips relative to each other so that there is no shared cross section.