Eye-AF disabled with some (not all) HDMI capture devices

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I'm trying to use a Sony Alpha A6100 with a BlackMagic Intensity Pro 4K capture device, and having problems with Eye-AF getting disabled.

For background, when I use the same camera with an Epiphan AV.IO 4K capture device, it works fine.  Specifically, I go into the Setup3 > 4K Output Select menu, select "HDMI ONLY (30p)", and get 2160p29.97 output (so not actually 30, but good enough since I know what I'm getting) with Eye AF working great.  Same thing happens with an Elgato Camlink 4K capture device.

By contrast, when I plug the camera into the BlackMagic capture device, several things are different.  First, the "4K Output Select" menu is unavailable.  Second, on my screen I get a blinking "STBY" icon, as if it weren't plugged into an HDMI device. Finally, of course, there is no Eye AF, which is unfortunate since the improved AF was the reason I got this camera in the first place.  I can confirm that I am getting 2160p29.97 output from the camera, though.

There are several reasons why I would prefer to use the camera with the BlackMagic Intensity Pro card (in addition to the fact that I paid $200 for the card and own it).  For starters, because the card is PCIe, it can do YUV 4:2:2 instead of 4:2:0.  Second, in my experience the USB capture can be not quite as reliable just because USB is a bit flaky (I tested 5 different PCIe USB cards before finding one that seems to work okay with 4K capture).

Can anyone thing of why the camera would not give me access to the "4K Output Select" menu with some capture devices, or if there's some other way of forcing Eye AF in my situation?  Thanks.