After an hour of research, trial, and error, I finally figured out how to get Con Emu to remember directories for tabs using Git Bash. When I first started using Con Emu, I swear it used to do this by default, but I guess updates in Con Emu or the Git Bash shell broke this feature. Either way, I've found a solution that worked for me to get this feature back.
According to the documentation, Con Emu is not able to monitor the changes in directory in the shell. As a result, we have to tell Git Bash to store the working directory, as it states here in the docs.
Open your text editor, or the terminal, and create a file in ~/ called .bash_profile and enter the contents which the Con Emu documentation states, which is shown below. After creating this file and entering the contents below, it should start remembering the directories.
PROMPT_COMMAND='ConEmuC -StoreCWD'
The file is .bash_profile, and not .bashrc because apparently when Git Bash is run with the --login flag, it reads the settings from .bash_profile and not .bashrc. The place I found this was a reply to a Stackoverflow question.
Note:
~/is an alias for the home directory of your user, e.g.C:\Users\{yourusername}, so the file's location would beC:\Users\{yourusername}\.bash_profile.