Edit: This was all wrong - my brain had the conversion direction reversed.

IGNORE ME!! (See new top-level reply.)

tl;dr; DisplayPort and HDMI speak "different languages", you'd need an active adapter like this random one that turned up in an Amazon search

The problem is that HDMI started out as single-link DVI over a different plug, then later transitioned into its own higher-speed-single-link DVI protocol. HDMI doesn't natively support dual-link. That's why HDMI was stuck at 1080p for a while - that was all single-link DVI could handle; that's what HDMI really was. When HDMI added >1080p support, it stayed single-link DVI, just sped up.

DisplayPort is its own separate "language". It never had a "dual link" situation, it just sped up the signal over the same number of wires.

Conventional "DisplayPort to <connector> adapters" (DisplayPort to HDMI, mini DisplayPort to VGA, whatever,) don't actually change the signal in the adapter. The adapter tells the video source "hey, instead of sending a proper DisplayPort signal, send me an HDMI signal" or "send me a VGA signal". The video port on the source (computer) then CHANGES WHAT DATA GOES OUT OVER WHAT PINS - so instead of sending digital DisplayPort data, it may be sending analog RGB data for a VGA adapter. Or single-link DVI data for a DVI (or HDMI!) adapter. The pins may not even match up with the proper "DisplayPort data" pins.

The mini DisplayPort to Dual Link DVI adapter turns dual-link DVI into high speed DisplayPort. That's it. It outputs a DisplayPort data signal only.

Adapting to any form of HDMI won't produce a usable signal. When using an adapter, the video source has to know to send a different signal. The Dual Link DVI to DisplayPort adapter doesn't know how to do that. It can ONLY feed DisplayPort. And because HDMI is a fundamentally different "language" than DisplayPort, you won't get it to work with a simple "passive" adapter that expects the video source to recognize and change what data it's sending. Thus you need an "active" adapter like I linked in the tl;dr which actually modifies the signal. (And I'll be honest, that was just the first Amazon result - and it seems WAY too cheap. I would expect an active adapter to cost similar to what the Dual Link DVI to miniDP adapter did when new.)

I use an hdmi extension cable with this displayport adapter and a usb 3 extension cable with this powered usb 3.0 hub. It's worked out really good, the powered hub solved the audio static issue, I got the recommendation here, I think.