Please note that HDMI-to-DVI and DVI-to-HDMI aren't the same thing. You note it correctly in your subject, but then say HDMI > DVI in your text (the signal does not travel from the monitor to the GPU). This is important because these active conversions are one way only.
Unfortunately, you need DVI dual-link to HDMI for this refresh rate. Which is a major problem; none of the adapters you'll find that say DVI dual-link to HDMI are active adapters, just passive. I'm unaware of an active adapter for this particular conversion.