To indicate that you are not at all concerned about something you can say I couldn’t care less.
I couldn’t care less if my old car broke down. I’ve been planning to buy a new one for some time now.
If his girlfriend left him, he couldn’t care less. He has found out that she is not his type.
So the phrase I couldn’t care less means that you don’t care at all.
Therefore it seems strange to hear I could care less, which has grown in use, particularly in American English.
He was so tired that he could care less if the roof fell down on him.
To me this indicates that he actually has some concern left, so the statement is actually illogical; it implies that he still cares, that he still has worries. As a copyeditor I recommend that you stick to the original version with couldn’t.