It now looks like the Immirzi parameter has a critical value which is determined numbertheoretically, not by the Bekenstein-Hawking S = A/4. It used to be that with LQG the IP appeared multiplicatively in S = A/4 so that the value of the IP had to be ADJUSTED to get the S = A/4 equation right. But this appears to be no longer the case. The B-H equation (S = A/4 + other terms ) has been derived in LQG without the Immirzi getting in multiplicatively in the leading term. So it is not forced or restricted. Interestingly however it gets a critical value from other considerations which happens to be around 0.274. This comes out in two recent papers, by Ghosh Perez and by Mitra. I mentioned this in another thread: The Immirzi comes into the picture when you add a purely quantum correction term involving an index of refinement N. N is the number of spinnetwork links that pass out thru the BH horizon. So if you REFINE the spinnetwork by adding nodes and links to it then intuitively you can be increasing N and adding a kind of quantum hair to the quantum BH state. The Immirzi determines the chemical potential the energy decrease associated with adding one more puncture. (Intuitively, dividing the area and curvature of the horizon up finer and finer.) I will go get the links to the recent papers by Ghosh Perez and by Mitra.