How do I set the /bin/sh prompt?

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I want to set my shell (set to /bin/sh) prompt to a bare $. I tried this:

sh-4.4$ cat ~/.profile
export PS1='$ '

However, as can be seen, the prompt remains unchanged. It works when I source .profile:

sh-4.4$ . ~/.profile
$

I've also tried adding the export to .bash_profile and .bashrc, because who knows, but that didn't change the prompt either. From what I've read, /bin/sh is implemented by bash on Fedora, and invoked as such, it should (only) source .profile from the home directory.

What should I do to permanently set my prompt to $?