Go to Preferences->Profiles->Default and find the option over on the right for "Working Directory". Then set it to "Reuse previous session's directory".
Then for me Cmd+D and Cmd+Shift+D (split with current profile) gives the same current directory.
Then for me Cmd+D and Cmd+Shift+D (split with current profile) gives the same current directory.
I want new windows (which I rarely use) and new tabs to be my home directory, but split panes to be the directory I was in when I split. Works wonders.