Literally, the word core refers to things inside a 3D structure (fruit, planets, etc). Figuratively, core has the notion of the thing or value etc that is 'central' or 'most important'. When used in the figurative sense (and in relation to a plurality of individuals, not just one individual), the term is conventionally used when there is one value or principle or 'heart' etc that is common to the whole group. In a sense, it wouldn't be 'core' if there were more than one of those things.
If core is used as a modifier (e.g. core activities - see below), then the word core is properly used in its base form, regardless of the number of (say) activities are involved.
core noun
The part of something that is central to its existence or character.
‘the plan has the interests of children at its core’
(as modifier) ‘managers can concentrate on their core activities’
- ODO
I'm not familiar with how WordHippo is set up. In particular, I don't know whether this is crowd-sourced information at one extreme or whether it is curated by a panel of experts at the other. The entry you link to has no examples of what it means by "the plural form will also be core", so I can't tell you what they were referring to. At a guess, they might have been referring to the 'modifier' sense in the dictionary example above.
The Ngram link you provided (and clicking through to "to their core") does have examples, though, so I'll comment on a few here:
Many people had found their refuge here; many people had connected to their heart and to their core here.
- Four Sides to the Core
The terms heart and core are used in the uncountable sense of 'basic value'. You can compare this with, say, "found their love" (uncountable) and contrast it with, say, "found their hobbies" (countable).
I started a journey toward stripping the standards to get to their core.
- The Core Deconstructed
This uses the term core in the 'collective most important bit' sense, hence the use of the singular. It can alternatively be read as a figurative use, which justifies using the base case.
Following this line of thinking, Chauncey Billups and Chris Bosh are both exceptional NBA basketball players, but they have unique, individualized needs relating to their core functionality.
- Conditioning to the Core
This uses the term core in the modifier sense (one noun modifying another), which justifies using the base case.
Note that the term core is not used in a plural sense in any of the above examples.
The "to their cores" click-through from your Ngram link does show some examples of cores used in the plural sense. Glancing at a few examples, it seems that in each case they refer to different 'cores' (e.g. one 'main thing' per company, not a unifying 'main thing' for the collection of companies), whether literal or figurative. The chart from your Ngram link shows that this usage appears less often than the singular or base case "to their core" (at least, in Ngram's database).
In any case, contrary to WordHippo's assertion, the examples above support the notion that the term core isn't used in the 'singular' to indicate the plural.
