EDIT (to add how it looks like it works to me): There are two headtubes, and two steerer tubes. The linkage bars are attached at pivot points to the headtubes, presumably just inside the bearing races so they dont' interfere with turning. The steerer tubes would be connected just like remote steering on many recumbents (including my CrazyBike2), which is what that third linkage on top is for. Turning the handlebars on the rear steerer tube pushes against the linkage which pushes against the front steerer tube's pivot point which turns the front wheel.
ORIGINAL:
The only potentially big issue I see with it is that all of the torsional loads of frame twisting are now no longer carried by the strong headtube-to-top&downtube welds, but instead are carried by the eight joints pinning the four linkage bars between the two headtubes. That's a lot of force to carry at such a tiny point.
If the points are beefed up then you need thick and/or strong metal for those linkage bars, so they themselves can't twist.
Otherwise you risk the front fork swaying side to side (not in steering) as you ride over a varying-angle surface, or pulling to the left in a righthand turn, and vice-versa.
It does look like a nice idea, but you may need to do it with dual-tubes on either side of the actual pivot as some of the photos show, rather than the much simpler method you have in the drawing (which I like, but am not sure it will hold up).
Another idea you could try that is pretty simple is to actually put the spring inside the headtube itself. I found this idea on the WISIL site originally, and am going to implement a version of it on CrazyBike2 at some point, to lighten the bike by eliminating the very heavy twin-shock front fork, using only a standard aero-tube ten-speed style fork plus the minimal spring/buffer/bearing/etc parts. Since it's a road bike, it doesn't *really* need the super-duper mountain bike suspension.
http://www.wisil.recumbents.com/wisil/headshock/headshock.htm
