The manual for your SDI to HDMI 6G converter states:
If your converter detects an HD monitor or TV connected to the HDMI output and has Ultra HD connected to the SDI input, the Ultra HD source will be automatically down converted so you can view the Ultra HD source on an HD monitor.
As your monitor does not support UHD resolutions, it will likely identify itself as HD, which would cause your converter to down-convert to 1080 either way. Besides, the product page doesn't list any 2K resolutions on the HDMI resolutions list, so it might down-convert those always. You best check with Blackmagic directly. I am not that familiar with HDMI, it might simply not include non-TV resolutions.
Lastly, since you wrote DCI, I assume we're talking 24fps? The only resolution listed as 24 Hz is 1080p. So even if you found a way to place your 2K image in a 2560x1440 raster and feed that to your monitor, it might not like 24Hz in that resolution.
So either contact BM and Eizo and ask for clarification or just call it a day and monitor in HD.