There is a conventional self, so is anything conventionally permanent? Answer from any reputable tradition.
The Buddha mentions three eternalist doctrines that exemplify conventional permanence. Let's skip the first two since they are weaker variations of this third one:
DN28:15.18: Furthermore, some ascetic or brahmin—by dint of keen, resolute, committed, and diligent effort, and right focus—experiences an immersion of the heart of such a kind that they recollect their past lives for as many as forty eons of the expansion and contraction of the cosmos, with features and details. They say, ‘I know that in the past the cosmos expanded or contracted. I don’t know whether in the future the cosmos will expand or contract.
This third conventional doctrine leaps to the following conclusion:
DN28:15.25:The self and the cosmos are eternal, barren, steady as a mountain peak, standing firm like a pillar. They remain the same for all eternity, while these sentient beings wander and transmigrate and pass away and rearise.’ This is the third eternalist doctrine. This is unsurpassable when it comes to eternalist doctrines.
In DN1, the Buddha explains the limitations of these very impressive conventional views:
DN1:1.36.1: The Realized One understands this: ‘If you hold on to and attach to these grounds for views it leads to such and such a destiny in the next life.’
And then the Buddha turns away from convention and goes beyond:
DN1:1.36.3: He understands this, and what goes beyond this. Yet since he does not misapprehend that understanding, he has realized extinguishment within himself. Having truly understood the origin, ending, gratification, drawback, and escape from feelings, the Realized One is freed through not grasping.
To paraphrase, "you can come back as much as you like, but why?"
Arahants and non-returners do not return and are not seen by humans or devas after their body breaks up. Conventional permanence seeks return. That is why it is bound to suffering. Because a vital condition for suffering is rebirth.
SN12.23:4.3: I say that suffering has a vital condition. And what is it? You should say: ‘Rebirth.’
There is nothing one should take as self as whatever you take as a self it will case unsatisfactoriness and is not permanent and not in one's control.
A normal worldling will consider conventionally that here is a self and it is permanent and it is satisfactory and it is in one's control. But it is not permanent and not in one's control and is not satisfactory.