They've given you the API spec.
So use that to build your own solution? (or pay for the one SmartBear provide)
That's what I did with TFS.
The part about direct export from Jenkins refers to JUnit/XUnit, Nunit, and TestNG again. So unless you can somehow force the TestComplete MHT file to be parsed into the same format as them (I have no idea where you are getting the idea it needs to be a PDF? A PDF file makes zero sense for this sort of work and the HP site states XML - as I mentioned above) and then hope you force it to treat it as if it was a JUnit/XUnit, Nunit, and TestNG test.
But this all very "square peg in a round hole".
You have an API you can use. HP have given you the spec for it. TestComplete can make the HTTP calls to it. You have everything you need to build your own solution so it works how YOU want it.
Yes, it takes a little more work, but good solutions to complex problems are seldom handed to you on a plate. For free.
SmartBear do offer a solution. But you seem to not want to use it for some reason? (Cost? I have no idea how much the license for the plugin is ...)