Sound transfer mitigation between Upstairs/Downstairs Apartments

[Will Adams Jr]

Hello!

Background:
We are finishing up the renovations on a SFH home with a split level design, in which we converted to an Up/Down duplex. At the onset, I knew that reducing sound transfer between floor/ceiling would be critical for this 1967 home. The steps we have taken thus far, which have reduced the sound transfer are:
1. Solid core door at the bottom of staircase, with spray foam seal in door jam 2. Weatherstrip on top and sides of door 3. rubber door sweep at the base of door 4. Blow in Sound+Thermal+Fire cellulose insulation between every floor joist

Problem:
There is still audible sound transfer of music/speech from the upstairs to the downstairs via the staircase that originally connected the two units at the Up entryway. I am looking for a solution to mitigate as much of this airborne sound as possible. the core problem is that of the sound transferring through the stairwell, ceiling insulations seems to be getting the job done.

Additional Information:
- I have access to the underside of the stairwell, but am unsure what material would provide, if any, sound reduction from the underside.

- The stairwell will not be used by the upstairs tenant, apart from entering onto the first landing.

Does anyone have a viable solution for preventing this? I am considering insulating from below the stairs, hanging a soundproof door cover, or any other solution that would not require me to rip the door out and fully separate the units, or lay down carpet on the stairs. 

Thank you!