Basayev's death is confirmed. Via chechnya-sl, a link to an YLE (Finland) newscast, which shows footage of the dead Chechen vice-president and guerrilla leader.
Update (July 12): there was obviously some confusion here, as the images shown on the Finnish broadcast were actually from 2000, and show Basayev being operated on after he stepped on a mine during the resistance's withdrawal from Grozny.
Update: Now Kavkaz Center has also confirmed the death, following the view that it was the result of a military accident, and not of a "successful special operation" by Russian or Russian-backed forces.
This is also the view of a political observer interviewed by Prague Watchdog's Umalt Chadayev - more later.