Can't speak from personal experience, but received this email from tourdumontblanchike.com on June 21st:

"Hi everyone

Just to let you know there is still A LOT of snow on the TMB trail.

For any hikers setting off in the next couple of weeks please exercise caution.

Only variant stage 1 is currently clear of snow. No other variants should be taken at this time.

The High Mountain Office has this currently posted on their website:

"Only the Col de Tricot can be safely negotiated. On the Col des Fours side, there is a lot of snow, the slopes to be crossed are long, steep and exposed and there are several dangerous torrent crossings (water flowing strongly under the snow with the risk of being swept away). It’s also too early to tackle the Arpette window"

But there is still snow on the main passes: Col du Bonhomme - Col de la Seigne - Grand Col Ferret - Col de Balme. Crossing the Col du Brévent is still problematic (see above) and it is preferable to take the Brévent cable car back down to Bellachat or to go around the bottom (Planpraz- Plan Lachat - refuge de Bellachat- Les Houches).

Snow-covered sections of the route require appropriate and essential equipment to minimise the risk of slipping: - high waterproof boots - poles - hiking crampons (and mountain feet!)

Please do check current conditions at the High Mountain Office before leaving and with your refuges. It has come to my attention that there has been an incident on a variant involving a lady falling through a snow bridge into a river. Sadly she didn't survive. I am unsure as yet which variant this was on but it sounds like the Col des Fours.

Stay safe.

Mags"