Control backlight through HDMI on BBB

I am working with a BeagleBone Black and an Ampire display (AM-1024600DTZQW). I would like to manage brightness with PWM, through HDMI and DDC protocol. Unfortunately, after much research, I am still stuck.

The HDMI of the BeagleBone uses i2c0 to communicate with the nxp TDA19988 controller. If I perform i2cdetect, I get the following result.

So I am missing the DDC address at 0x37. Moreover, if I try ddccontrol, I get this:

However, I am able to read the edid with parse-edid.

My Ampire screen has been implemented with a firmware that supports DDC protocol. And it works fine when I connect it on my Linux PC.|952.8239113092422x541.1355657577515

Also, I had no problem working with my BBB and another display (Digital Display without DDC).

I don’t really have any more clues on how to solve this issue. Does anyone have an idea?