They'll be pumping neutral gas (probably N2 or argon) to purge the borehole. I would expect the ablated material to resolidify as fine particulate, which would get carried to the surface at ambient temperature and trapped in a filter.

A 20km borehole with a 10cm diameter is only ~157m3 of rock. If the dig takes four weeks (which would be unprecedentedly fast), you only need to purge ~64cm3/s, which is pretty trivial.

They'll need to do something like mix the dust into concrete. Silicate dust is a respiratory hazard. But that doesn't strike me as a significant hurdle.

Apparently that's what they intend. I wonder how you can keep 20km oven warm enough.