Inside this small and vexed question even smaller questions are struggling to get out.
The most detailed discussion to date appears to be
Alfredo R. Paloyo. 2011. When did we begin to spell “heteros*edasticity” correctly? Ruhr Economic Papers 0300.
(a reference I owe to @Andy here in Ten fold chat). I can't do justice to its dense and detailed discussion. What follows is more by nature of an executive summary, modulo a little whimsy.
Modern search facilities make it possible to be confident that homoscedastic(ity) and heteroscedastic(ity) are modern coinages introduced, explicitly or implicitly, by the British statistician Karl Pearson in 1905. (Pearson ranged widely over several disciplines, but in the second half of his life his work was firmly centred on statistics.)
Modifying c to k raises absolutely no statistical issue. The idea is at its simplest that the Greek root being used includes the letter kappa ($\kappa$), whose direct equivalent in English is k, and so that k is the correct spelling.
However, as others have done elsewhere, we note that this suggestion was made particularly by J.H. McCulloch in the journal Econometrica, a journal which failed to follow the same logic by renaming itself Econometrika, nay Ekonometrika. (The roots behind "economics" are also Greek, including the word oikos. Ecologists will want to add that there is a journal Oikos even though, once again, ecology did not call itself oikology.)
Further, it is remarkable that Karl Pearson was no hater of k, as he changed his own name from Carl to Karl and named his own journal Biometrika, in full and conscious recognition of the original Greek words he used when devising that name.
The root question then is purely one of language, and of how faithful it is proper to be to the original words behind a coinage. If you follow up the McCulloch reference, the discussion turns to whether such words came into English directly or via other languages, and so hinges on criteria that may appear to many readers as arbitrary if not arcane. (Note that criteria is another word of Greek origin that escaped the k treatment.) Most language authorities now acknowledge that present spelling can owe much to historical accidents and that any long-established usage eventually can over-turn logic (or more precisely etymology). In total, there is plenty of scope here for scepticism (or skepticism).
In terms of tribal or other preferences, it is my impression that
Econometric usage seems to be shifting towards the k form. The McCulloch paper had an effect, indirectly if not directly.
British English seems to make more use of c forms over k forms than does American English. The form sceptic is standard in British spelling, for example.
All puns and wordplay here should be considered intentional even when accidental.