This and more (saving of watch window variables) should now be possible with a recent commit on github.
Essentially, prior to exit, you can have something so:
nnoremap <leader>d1 :call vimspector#WriteSessionFile(".vim/debuggingsession.txt")<CR>
Then, the next time you are in, you call:
nnoremap <leader>d2 :call vimspector#...