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What was Muchukunda's next birth?

Muchukunda was an ancient solar dynasty king who was the son of Vishnu's incarnation Mandhata, whom I discuss here and the brother of Vishnu's devotee Ambarisha I discuss here. Before the birth of Kartikeya, Muchukunda briefly served as the general of the gods in their battle against the demon Tarakasura. But similar to the story of Kakudmi, during that brief time a great deal of time had passed on Earth, during which all his family, friends, and citizens had perished. Since everyone he knew was gone, Muchukunda went inside a mountain cave and slept for eons. Finally he was awoken during the time of Krishna by the e Greek king Kalayavana, who along with Jarasandha had just attacked Mathura (leading to the Yadavas going to Dwaraka. Angered by being awoken, Muchukunda burnt Kalayavana to ashes.

That's all that most people know about Muchukunda, but I'm interested in what happened after that. This chapter of the Srimad Bhagavatam describes how after Kalayavana was killed, Vishnu appeared before Muchukunda and granted him the following boon:

O emperor, great ruler, your mind is pure and potent. Though I enticed You with benedictions, your mind was not overcome by material desires. Understand that I enticed you with benedictions just to prove that you would not be deceived. The intelligence of My unalloyed devotees is never diverted by material blessings. The minds of nondevotees who engage in such practices as prāṇāyama are not fully cleansed of material desires. Thus, O King, material desires are again seen to arise in their minds. Wander this earth at will, with your mind fixed on Me. May you always possess such unfailing devotion for Me. Because you followed the principles of a kṣatriya, you killed living beings while hunting and performing other duties. You must vanquish the sins thus incurred by carefully executing penances while remaining surrendered to Me. O King, in your very next life you will become an excellent brāhmaṇa, the greatest well-wisher of all creatures, and certainly come to Me alone.

I'm interested in the part in bold. My question is, what was the next birth of Muchukunda as an "excellent Brahmana"? Did this birth already happen, or is it going to happen in future?

The next chapter of the Srimad Bhagavatam says that Muchukunda subsequently went to Badarikashrama to engage in Tapasya, but it doesn't say when or how he died. In any case, there are claims that Prahlada was reborn as the Dvaita philosopher Raghavendra, that Arjuna was reborn as the Shaivite poet Kannappa, etc. Are there any claims that some Kali Yuga figure was a rebirth of Muchukunda?