No significant progress. But there's some still hope.
On 10th September, I received a call from a lady (Apparently their underwriting team) who asked me what my profession is, and why my home and office addresses are the same. She was in hurry and disconnected the call without letting me ask further questions. But thankfully, that landline number could be dialed back and the same person picked it up. I asked whether it would hit CIBIL again or not, she just replied, "No" and hung up again herself.
The next day, 11th September, the Customer Service representative from the "Credit Cards Head" team, who replied to my email on 5th September, called me up (There were no communications from them between 5th and 11th September). She informed me that she would raise a service request for the limit enhancement, and assured me I would get a resolution within 7-10 working days. However, when I asked her if it would hit CIBIL again or not (I wanted to be double-sure), she said she will ask her team and get back to me later, and only after a confirmation she would proceed with the SR creation if they confirm it won't hit CIBIL.
And since 11th September, I'm still awaiting her callback.
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In case anyone else is interested what this case was about: When I was applying for their cards on their website, right after I consented them to access my CIBIL Report and accepted their T&Cs, they showed me a credit limit of ₹1.5L, which is why I proceeded ahead with the application. However, when I received the card, I was stunned to notice that the actual credit limit was ₹75K instead, which was why I escalated this issue from Level 2 to their highest level of escalations one at a time.