Umm, just been out for the week and there is A LOT of snow.
Walked up to the Tre la Tete hut and spent a day following a ridge on the Aguille Rouge and there was snow from around 1400m. We were the only people out. Even the guides weren't taking anyone out.
Office de Haute Monagne were telling people not to bother trying the TMB and a few parties we spoke who came out to do it didn't. The answer to all our questions regarding (climbing) routes was 'not possible without skis or snow shoes'. Also high avalanche risk (although I assume this is not such an issue for TMB?).
Weather supposed to be picking up so might be gone/consolidated by the time you get out there?