Well, a bit more circumstantial evidence in support... I located a work on writing, Faulmann's
Illustrirte Geschichte der Schrift, which covers Kufic script, the original calligraphic style used for transmitting the Qur'an. P. 416 gives the 4 forms of the various letters, and you can see that ك has essentially the single form as it does in Persian. Also p. 39 of
this article on Maghribi script (likewise archaic but not as archaic) has more uniform shapes to kaf.
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Illustrirte Geschichte, p. 420 discusses the features of Naskh, which is closer to Ruq`a that we encounter nowadays, and says that "
k is more distinguished from
l with an 'inscribed' ء", etc. indicating that kaf was elaborated in this script.