Hi Dave,
My situation isn't entirely the same (k-5 and old Sigma 170-500 I got about a month ago), but the behaviour is the same indeed. If the AF starts hunting, it won't start again once it has reached the close end of the focusing range.
I'm not quite sure why, but technically the f/6.3 (f/6.7 in real life) maximum aperture is smaller that what my camera's manual specifies for the AF to work (f/5.6), so maybe that plays a role. I've noticed that when I simply attach a 1.7x rear converter to my 70-300 mm f/4-5.6 zoom, the AF continues to work, and my guess is that is because the camera still reads this combo as f/5.6. But that is pure guessing on my part.
Another thing may be that once you focus on something quite far, it appears to fall between the AF sensor points. I just focused on a large building several hundreds of meters away, and that works without problems.
Essentially, I'm not sure if the problem lies with the camera or with the lens. I'm guessing the camera, given how my old version of the lens has no active components.
I have to say, most of the time I use the thing in MF mode anyway, simply because of the fact that my typical subjects tend to hide between twigs, which is AF nightmare. Only if something is close or rather big, with no obstacles between it and the camera I use AF. Invariably those pics are a tad sharper than when I use MF (maybe I simply need glasses), so I feel your pain
.