Sri Sunil ji,
I will share with you what Sri Chandrasekharendra Saraswathi of the Kanchi matam has said about the above topic. Below are a few excerpts verbatim from the "Voice of God" book Vol 5 (pages 708-737), which is a translation of the book "Deivatthin Kural" that is a collection of his various speeches.
On Acharya and the five spatika lingams:
"... he went to Kailas to have dharshan of Parvathi and Parameshwara. It is said that he went there by Yogashakti. ...
... Parameshwara gave Acharya five spatika lingams. This has been mentioned both in Siva Rahasyam and Markandeya Samhithai which are part of Itihasa-Puranam. In Siva Rahasyam it has been stated that this incident happened in Kasi. But the lingams have been referred to as Kailasa Lingam. The names of the lingas have been mentioned in both identically as Yoga Lingam, Bhoga Lingam, Vara Lingam, Mukthi Lingam, and Moksha Lingam. In Markandeya Samhithai it has also been mentioned where they were installed. Yoga Lingam was installed in the Mutt at Kanchi, Bhoga Lingam in the Mutta at Srungeri, Vara Lingam in Nilakanta Kshetram in Nepal, Mukthi Lingam in Kedarnath and Moksha Lingam in Chidambaram. ...
... In accordance with Parameshwara's command Acharya installed four lingams in Srungeri, Kedar, Nilakantam, and Kanchi. When he was nearing the time for 'siddhi', he gave the 'moksha lingam' to Sureshwaracharya and told him to install it in Chidambaram. This is what is found in Ananda Giriyam. ...
... There are historical narratives which talk about his having installed the Yoga Lingam which he had received from Parameshwara in Kailas as Chandramoulishwara in the (Kanchi) Mutt. About this Yoga Lingam there is reference even in Sri Harsha's Naishadham."
On Acharya's Videha Mukthi in Kanchi:
" ... Almost all books speak of Acharya having been in Kanchipuram. Both in Ananda Giriyam, (Srungeri) Guruvamsa Kavayam it has been mentioned that it was from Kanchi he sent out his six sishyas all over and established the Shanmatha. Sankaraabhyudham, Govindanaadhiyam, Chitvilaasiyam and Guru Rathnamaala say that Acharya ascended the Sarvajna Pitam. From Sushama, 'Bruhath Sankara' and 'Praachina Sankara Vijayam' books also this fact is known. Although according to Vyasaachaliyam he ascended the Sarvajna Pitam in Kashmir the trend of his life history shows that he had lived in Kanchi. By way of clearing this confusion, Kanchi region itself has the name 'Kashmiram' as made clear in Govindanaadhiyam. ...
... In Siva Rahasya, Markandeya Samhithai which are Itihasa Puranas and in some Sankara Vijayam books and other books pertaining to Acharya it is said that Acharya ended his avathaaram and attained siddhi in Kanchi. There are also some Sankara Vijayam books which say that it was in Kedarnath, Badrinath he attained 'siddhi'.
It is difficult to appreciate differing opinions being expressed concerning the history of Acharya who had come only for bringing about unity. He did not write his autobiography or leave a 'will'. But those who came after him drew their own conclusions based on different evidences. The basic truth is that all of us are his bhakthas, sishyas and children. More important than conducting any research is that all of us should have bhakthi to him and love devoid of even a tinge of enmity. If anything does harm to this all the research can be withdrawn. (laughing) If some one asks 'which is the place where Acharya attained siddhi?', just as we say about 'siddha purushas' that one person had attained 'samadhi' in three or four places we can also say about Acharya!
The belief is that Acharya attained mukthi in Kanchipuram which is a 'mukthi kshethra'. What is mukthi for him? He was the very form of mukthi. He was always 'Jivan Muktha'. Even so one day he had given up his mortal coils and disappeared from our sight. I am referring to Videha Mukthi. In some of the books it is mentioned that he attained such a mukthi in front of Kamakshi. In Bruhas Sankara Vijayam it is said : Devyaah purah parathare purushevililye - in front of Ambal he became one with 'Brahma tatva'. In Prachina Sankara Vijayam also it has been mentioned in the same manner:
Kamakshyaat savithe sa jaathu nivisan unmuktha loka spruho
Dheham swam vyapahaaya dhehya sukamam dhaama prabethe param
Sa jathu kamakshyaah savithe nivisan - one day he went to Kamashi temple and was standing near Ambal.
Unmuktha loka spruha - At such a time he had given up all attachment to the world.
Dheham swam vyapahaaya - standing near Ambal he gave up his body.
Dhaama prabethe param - He returned to his own highest place. What is that highest place? It is being Brahma Swarupam.
In Keraliya Sankara Vijayam also this has been said and in addition it is said 'even after attaining the supreme state (Paramapadham) of limitless Jyothi, the very form of bliss, Sri Sankaracharya contines to remain in the temple of Ambal in Kanchipuram as 'Moksha Dhatha'. It appears that the reference here is to Acharya's idol in Kamakshi temple. ..."
Om tat sat 🙏