Looks like frame 17764 from the spreadsheet + 2 frames lost on the axe grab = 17766 or 4:59.939, but please double check my timing.

Please see the YouTube video description for a more detailed writeup.

Here is also a link to my first deathless completion of 5:03 (with this one being my second deathless run):

Played on a model 1 US NES with an original authentic US Super Mario Bros./Duck Hunt combination game cartridge.