I don't what the research is in question, but I absolutely think that political extremism, particularly in the context of two party systems, can quantified in at least a semi-meaningful way.
Create a list of policies. Poll a large equally split group of registered voters from either party. Assign each policy an extremism index based on some combination of the amount of cross party support, and the percentage of total support for each.
Now you can use that list to make a reasonable assessment of political extremism based on which of the policies an individual supports.
It does if and only if that's what you reduce extremism down to.
There's enough there to assert a quadratic relationship (in a given regime) with _their particular phenomenological measurement_ of extremism.
In the physical sciences we can often get away with ignoring the difference between what a measurement procedure outputs, and what it is measuring. But this only because we have high degrees of repeatability, and concordance with other ways of measurement. Neither has been shown to apply here.
The reporting of this is made worse by the use of a common term, rather than the diagnostic tool. Yes, there are common meanings of "energy", "momentum", and so forth, but people understand that they mean very specific technical definitions when a physicist uses them.
Extremism has no natural quantification. This statement is nonsense.