That is what the projections from the Congressional Budget Office (CBO) imply. And, just to be clear, this is the additional waste that CBO projects would result from running the program through private insurers. The figure does not include roughly $5 trillion in expenses that would be transferred from the government to insurers.
The $34 trillion figure comes to $110,000 for every man, woman, and child in the country. It is almost 7 times the projected Social Security shortfall. While the Post has devoted endless space (in both its news and opinion sections) hyping the need to fix Social Security, it has never once mentioned this much larger cost to the country projected by CBO. Instead its articles on the Republican Medicare plan have been entirely of the he said/she said variety and discussions of its political prospects.
A serious paper would discuss the substance. The Post's readers don't have more time to look into this issue than its reporters.