I have checked that in most of the threads considering the scenario of a possible Hellenistic India it has been discussed how complicated could it be for the Hellenistic culture to supersede, even if partially, the well established Indian cultures and form a real Hellenistic state which could prosper long enough in order to become a powerhouse that could dominate South Asia and the Indian Ocean. Certainly, India was too big, culturally diverse and demographically strong by the era in order to be affected enough by an eventual Hellenistic dominance.
But let's consider that Indo-Greeks managed to establish a safe and more manageable (smaller) stronghold in the area, in the island of Taprobane (today Sri Lanka). To convert all or most of the India into the Hellenistic culture is quite ASB, but Taprobane could be a more feasible target, as it was way smaller, but already known for their interesting commerical goods like ivory and spices, so this would allow to establish a prosper economy. The situation is also good in order to establish seafaring routes to places like Madagascar or West Australia, opening them to further settlement during the following centuries.
It might seem that Greeks successfully settling Taprobane is quite far-fetched, but this could come if we think on a gradual scenario: first the Indo-Greeks succeeds in expand their original stronghold in modern Pakistan and later expand a bit to the western coast of the Indian subcontinent (Mumbai). As the Greeks already knew the existence and wealth of Taprobane, they can first establish trading routes from their controlled Indian lands and start a slow introduction of their culture in the island. As the Indo-Greek state(s) in northwestern India would likely fall sooner or later, there could be a final migration of Indo-Greeks fleeing the subcontinent for the safety of the island. They might establish some kind of commercial and cultural elite there that could eventually jump into political dominance, which would potentially be easier to maintain than anywhere in continental India.
But let's consider that Indo-Greeks managed to establish a safe and more manageable (smaller) stronghold in the area, in the island of Taprobane (today Sri Lanka). To convert all or most of the India into the Hellenistic culture is quite ASB, but Taprobane could be a more feasible target, as it was way smaller, but already known for their interesting commerical goods like ivory and spices, so this would allow to establish a prosper economy. The situation is also good in order to establish seafaring routes to places like Madagascar or West Australia, opening them to further settlement during the following centuries.
It might seem that Greeks successfully settling Taprobane is quite far-fetched, but this could come if we think on a gradual scenario: first the Indo-Greeks succeeds in expand their original stronghold in modern Pakistan and later expand a bit to the western coast of the Indian subcontinent (Mumbai). As the Greeks already knew the existence and wealth of Taprobane, they can first establish trading routes from their controlled Indian lands and start a slow introduction of their culture in the island. As the Indo-Greek state(s) in northwestern India would likely fall sooner or later, there could be a final migration of Indo-Greeks fleeing the subcontinent for the safety of the island. They might establish some kind of commercial and cultural elite there that could eventually jump into political dominance, which would potentially be easier to maintain than anywhere in continental India.