Currently, the best possible answer, although not a verifiably correct answer, is "over 4400K" or whatever the exact value determined in the research mentioned from the article you provided is. It actually does describe the "Theoretical minimum temperature required to melt any material” that you're looking for, at least as best as any human currently can.
In theory there is a temperature you could prove no bonds can be maintained between multiple elements, which we already can prove is higher than the melting point of any individual element. However, that temperature may also be far higher than the minimum needed to melt any compound that could exist (or, a maximum melting point).
There also (sh)/(c)ould be undiscovered compounds that take a higher temperature to melt than this newly("newly", 6 yrs ago) discovered material you linked. (and I have not researched to confirm this isn't the case) - but this is an NP-Hard problem. Anyone here that can solve this, I suggest you check out the Clay Mathematics Institute before you post anything...
So, the question can't be answered any better than confirmation from the scientific community of a newly discovered compound with a new record for melting point, until we, as a species, figure out timely solutions to NP-hard problems. This doesn’t preclude studying known high melting point substances to narrow the possible fields and more rapidly develop a new compound with a higher melting point - but it is still not possible to prove it’s the highest melting point.
That any one compound has the maximum melting point cannot (currently) be proven, this makes it impossible to define the maximum melting point.