mtDNA K may well have come to an Anatolian LGM refuge along with I2
I would think I2 came to Anatolia after its formation, 27.5 ka, but before its TMRCA 21.9 ka
https://www.yfull.com/tree/I2/
No I2 has been observed in mesolithic Europe prior to the dispersal of the Villabruna clade, so I would suspect it stayed in Anatolia all that time.
Note that the Villabrunans arrived in Europe with microliths made with 'burin' technique and probably also with bow and arrow, something that appeared in the Kabaran in the Levant and in the Zarzian in the Zagros Mts during LGM.
mtDNA K3 was in Satsublia HG, but K1a was in Boncuklu PPN and Tepecik PN, K1c was in mesolithic Greece, while both K1a and K1b were in Barcin neolithic, all in association with G2a
the problem with mtDNA is that there is no reliable dating of the TMRCA's in the pedigree, we don't know when K or K1 originated or split
the recent paper on Indian DNA
http://www.eupedia.com/forum/threads/34112-Genetic-history-of-the-Indiancattluontinent?highlight=Indian claims reliable dating of the mtDNA pedigree based on a 2009 paper, but I have my doubts