HPV. How do viruses persist outside the body?
bob1's answer is good, but I was already working on this one so I'll go ahead and post the parts that aren't redundant:
What does it mean for a virion to lose infectiousness?
To be 'infectious' a  virus particle perform a set of tasks that result in more virus particles. A simplified list for HPV could be: attach to receptor on host cell &gt; enter cytoplasm &gt; enter nucleus &gt; make proteins &gt; copy DNA &gt; assemble new virions from protein and DNA &gt; exit cell. The first four tasks must be performed by proteins in the virion; if enough of these are degraded, the corresponding task cannot be completed, and the virion cannot infect a cell. This means that there are many different routes by which any particular virus might be inactivated. One of the references in the question suggests HPV can be inactivated by gunk from dead cells sticking to the otherwise-viable virions:

The greater inactivation of the virions in the extracts at 56 C may be due to denatured cellular protein, which visibly aggregated at 56 C and above, interfering with infection.

Viruses are not metabolically active on their own!
This means that damage cannot be repaired, and also that a functional protein cannot be 'reset' after it changes shape. To use the SARS-2 spike protein as an example, the spikes have energy stored in them like a mousetrap. If the transition from open to closed is triggered while the spike is bound to ACE2 on a host cell's membrane, the motion of the spike protein changing shape will bring the viral membrane in contact with the host membrane, fusion occurs, and the virion contents enter the host cytoplasm. If the transition is triggered before the spike is bound to a host receptor, that spike is 'dead'. SARS-2 virions have many spikes, but if all of them trigger prematurely, the virion as a whole becomes uninfectious. The transition from the high energy 'prefusion' state to the lower energy 'dead' or 'postfusion' state has to be energetically accessible in order to be triggered by the new host cell. This means that the risk of premature triggering is inescapable. Therefore, damage or premature transition of all copies of some viral protein that changes shape to allow cell entry is a likely candidate for inactivation for any viral species. This hasn't been investigated systematically, though, so there's no reference I can cite for this.
In papillomaviruses:
Unlike coronaviruses, HPV doesn't have a membrane that can fuse with the host's for cytoplasm entry. Instead, it is absorbed by endocytosis, then the capsid disassembles, and L2 disrupts the endosome membrane. The details of how this happens aren't known, but it's likely that L2, which is protected inside the capsid most of the time, becomes exposed by structural changes to L1 which occur in the endosomes. This suggests to me that degradation of L1 is the main cause of viral inactivation outside of the cell. That hypothetical structural change of L1 would need to be identified before you could start to investigate if it's present in inactivated particles.