This DTIC Report (PDF) shows that the conductivity of silicon continues to increase as temperatures increase from 500K to close to silicon's melting point at 1,687K:
This is due to the rapid increase in the number of free electrons and holes with increasing temperature.
As temperatures increase, phonon scattering increases and this reduces the mobility. The rate of increase of scattering with temperature is much less than the rate of increase in free electrons and holes with temperature. Hence, conductivity continues to increase as temperature rises.
Intrinsic vs extrinsic
The plot below, from the same report, shows how the conductivity of intrinsic silicon (samples 5 & 6) compare with doped (extrinsic) silicon (samples 1 and 2). Note that, unlike the plot above, the scale of the top axis on the plot below is in centigrade, not Kelvin:
At low temperatures, the doped silicon has higher conductivity than the intrinsic silicon semiconductor. As temperature increases, the conductivity of the doped Si drops slightly to due increased scattering. As temperatures increase further, the growth in thermal carriers becomes more important than the increase in scattering. As temperature increases further, the conductivity also increases.