The source for Sanjeev Sanyal's account is most likely Plutarch. In his Life of Anthony, Plutarch wrote:
Caesarion, who was said to be Cleopatra's son by Julius Caesar, was
sent by his mother, with much treasure, into India, by way of
Ethiopia. There Rhodon, another tutor like Theodorus, persuaded him to
go back, on the ground that Caesar invited him to take the kingdom.
Our sources are somewhat unclear but, putting together what evidence has survived, Michael Gray-Fow's 2014 article What to do with Caesarion in the Classical Society journal Greece & Rome, says that Cleopatra, when deciding in 30 BC where to flee,
… opted for India and organized a fleet on the Red Sea, but it was
destroyed by the Nabateans because their king Malchus (Maliku II)
wanted to demonstrate his value to Octavian…
Octavian approached Alexandria, Cleopatra decided that Caesarion’s only salvation lay in getting as
far away from Octavian as possible. For him alone she resurrected the
idea of flight to India.
Caesarion was sent off up the Nile with his tutor, Rhodon…Octavian
entered Alexandria on 1 August 30 BC. Just over a week later,
probably believing that Caesarion was safely on his way to India and
that Octavian might be more lenient towards him if she were dead,
Cleopatra also committed suicide.
It is doubtful whether Caesarion ever reached Myos Hormos or Berenice [Red Sea ports].
He certainly got no further if he did. Exactly what happened is
unclear, perhaps deliberately so. Both Dio and Suetonius agree that
Caesarion was overtaken in his flight, though Dio alone implies that
he was murdered at that point. Plutarch and Suetonius both claim that
Caesarion was brought back to Alexandria and killed there.
Gray-Fow cites numerous sources. Among the primary ones are Plutarch (c. AD 46 – AD 120), Livy (64 or 59 BC – AD 12 or 17), Cassius Dio (c. AD 155 – c. AD 235) and Suetonius (c. AD 69 – AD 122).