This ("cross-lot" connector pathways) is a topic I've gotten interested in lately, so I can offer a few pointers:

The city's interactive transportation GIS map includes some, but not all, such pathways in the city: https://experience.arcgis.com/experience/5730c9e150da44c4876c980b9be7aa1e. You need to enable the "sidewalk" and "non-motorized routes" layers to see them. If you can find the relevant pathway in that tool, it will tell you - to the best of the city's understanding, at least - whether it's privately or publicly owned.

After that, if a path shows as private or is not visible at all on that map, the next step is to look up development plans, and in particular, "site plans". You can search for an address in https://etrakit.a2gov.org/etrakit/Search/project.aspx to find plans for the last couple decades; prior to that, many documents are available on https://www.a2gov.org/departments/build-rent-inspect/building/Pages/archived-project-lookup.aspx.

This particular path is on property owned by Speedway, and it appears in the site plan that they submitted in 2012 to rebuild the station: https://etrakit.a2gov.org/etrakit/viewAttachment.aspx?Group=PROJECT&ActivityNo=SP11-037&key=SH%3a2011130303076656

That site plan specifically mentions that in the course of that project, they were going to reconstruct that pathway (Page 5 of the PDF, element E13). It's not clear to me that they actually did that - the extent to which it has heaved and crumbled makes me doubt it - that seems more than 10 years old? - but I'm certainly no expert on pavement longevity.

In any case, it is my understanding that private property owners are responsible for ongoing maintenance of elements within an approved site plan...