There is no hardware solution, short of some crazy* arrangement of mirrors to extend the actual optical distance. That's because perspective distortion solely related to distance, but there might be a software one, if you're able to throw a computer at the problem and able to accept some limitations.

For a computational approach, see this paper: Perspective-aware Manipulation of Portrait Photos (pdf link) — the basic idea is that you build a 3D model of the head and then use that to distort the image to match a photograph taken from a different distance. Unlike many computer graphics or computational photography papers, this one is quite readable by the layperson (e.g. me), although I admit to glossing over the actual hard part about the tensor model.

@Ian points out that there is an online demo. I ran your example through that, with this result:

This solution uses a single image; I'm not sure if that's part of your constraint. Having more information for doing this kind of sorcery in software is one of the uses for multiple simultaneous cameras — think some of the new smartphones.

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* If you're got a fixed location, it doesn't actually have to be so crazy. Place a large mirror pointing up at a 45° angle against the wall. Mount your camera near the ceiling. Keep the mirror clean!