I tried this. It was more than a few thousandths, but it worked. I removed a little metal from the shell holder, then ran the brass through the die. I could see the shiny ring going further down the case each time. 99 out of 100 dropped in the barrel. I'm happy.
This is why this reloading section exist! Congrats on obtaining a great result!
I had to do the same with .308 RCBS dies and Ruger 77V series.