It's TRUE.
Rechambering an existing .308 Winchester to .30-06 will leave a bulge in the .30-06 chamber below the shoulder. Not exactly what you want for reliable extraction. A proper conversion will require either setting the barrel back (if possible), or a new barrel.
The US Military experimented with firing .308 Winchester (7.62x51 NATO) ammunition in existing .30-06 rilfes in the late 1950's. The reason was to determine user safety if 7.62x51 was accidentally fired in M1 rifles, BAR, etc. It was found that there was no hazard to the firer because the wider shoulder of the .308 Win case gave complete obturation on firing. Whether this was by design, or by accident, I dunno.
I do know that the US Navy converted several thousand .30-06 M1 rifles to 7.62 NATO using a chamber "sleeve" and they required running a reamer into the chamber to make it accept 7.62 ammunition. I found the same thing in the two .30-06 rifles I converted to .308 Winchester.
HTH
John