The space-time of general relativity corresponds to effective space-time obtained by replacing the sheets of space-time with single region of Minkowski space with metric replaced with the sum of empty space Minkowski metric with the deviations of the induced metrics of the space-time sheets (effects of classical fields on space-time sheets on test particle sum up since it touches all the space-time sheets: linear superposition of fields is replaced with that for their effects).
SN1987A supernova provides first evidence for the presence of several space-time sheets. Neutrinos came as two bursts and before photons. The variation of effective light-velocity was of order Δ c/c ≈ 2× 10-9. Opera experiment claimed much larger variation of order: Δ c/c ≈ 10-5: unfortunately there was an error in the analysis of the experiments. Now Lubos has a posting about galactic blackhole Sagittarius A as neutrino factory. Chandra X-ray observatory and also Nustar and Swift Gamma-Ray Burst Mission detected some X-ray flares from Sagittarius A. 2-3 hours earlier IceCube detected high energy neutrinos by IceCube on the South Pole. As a good conservative Lubos of course denies the effect as he denies also climate warming. He did not however claim that experimenters are communists;-).
Could neutrinos arrive from galactic center? If they move with the same (actually somewhat lower) velocity than photons, this cannot be the case. The neutrinos did the same trick as SN1987A neutrinos and arrived 2-3 hours before the X-rays! What if one takes TGD seriously and estimates Δ c/c for this event? The result is Δ c/c ∼ (1.25-1.40 )×10-8 for 3 hours lapse using the estimate r= 25,900+/- 1,400 light years (see this). Δ c/c is by a factor 4 larger than for SN1987A at distance about 168,000 light years (see this). This distance is roughly 8 times longer. This would suggests that the smaller the space-time sheets the nearer the velocity of neutrinos is to its maximal value. For photons the reduction from the maximal signal velocity is larger.