Is the giant who carved the “Under Me” line Time?

Question

In The Silver Chair, Rilian, while enchanted, said:

Though under Earth and throneless now I be,
Yet, while I lived, all Earth was under me.

From which it is plain that some great king of the ancient giants, who lives¹ buried there, caused this boast to be cut in the stone over his sepulcher;...
_{-The Chronicles of Narnia: The Silver Chair, chapter 10}

And before, when they are traveling to the Underworld:

"That is old Father Time, who was once a king in Overland," said the Warden. "And now he has sunk down into the Deep Realm and lies dreaming of all the things that are done in the upper world. Many sink down, and few return to the sunlit lands. They say he will wake at the end of the world."
_{-The Chronicles of Narnia: The Silver Chair, chapter 10}

Now, Time fits the description of the scene that Rilian gave: He's a giant, he was a king, and he is located under the carving.

Was Father Time the one to have the message carved in the stone?

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_{¹ Yes, it says "lives buried" in my version.}

Answer 1

It appears so.

Since nobody else answered with an explanation of how everything fits, I decided to go ahead and post my own.

• Though under Earth and throneless now I be

This certainly applies - he is buried and has no crown.

"And now he has sunk down into the Deep Realm and lies dreaming of all the things that are done in the upper world. Many sink down, and few return to the sunlit lands. They say he will wake at the end of the world."
_{-The Chronicles of Narnia: The Silver Chair, chapter 10}

• Yet, while I lived, all Earth was under me.

If he was a king, then metaphorically, yes, he's above everyone else.

• From which it is plain that some great king of the ancient giants, who lives buried there, caused this boast to be cut in the stone over his sepulcher;

"That is old Father Time, who was once a king in Overland," said the Warden.
_{-The Chronicles of Narnia: The Silver Chair, chapter 10}

So he was king, and we see that he is a giant:

And here, filling almost the whole length of it, lay an enormous man fast asleep. He was far bigger than any of the giants, and his face was not like a giant's, but noble and beautiful.
_{-The Chronicles of Narnia: The Silver Chair, chapter 10}

Also - who else would it be?

Is there any other giant king that we know of buried right there?

It seems entirely possible that it was time who had it carved.

Answer 2

I think he is the most likely option. From reviewing the books and skimming, he seems to most definitely be the only giant located under the carving. However, he isn't the only king, but in my view, "who lives buried there" is the more important criterion and as the giant Time fulfills both, I think he is the most likely option.