The fate of infalling matter in the context of black holes is a topic of ongoing research and debate in theoretical physics. The current understanding based on general relativity suggests that once matter crosses the event horizon of a black hole, it is destined to reach the central singularity.
Time dilation is indeed a factor to consider near black holes, as gravitational forces can cause significant time dilation effects. However, it's important to note that time dilation affects both the infalling matter and the external observers outside the black hole. From the perspective of an external observer, the infalling matter appears to slow down as it approaches the event horizon, but it never quite reaches it. This is known as the "coordinate time" perspective.
However, from the perspective of the infalling matter itself, its own subjective experience of time is not affected. The matter will continue to fall towards the singularity and eventually reach it in a finite amount of proper time. This is known as the "proper time" perspective. In this frame of reference, the matter crosses the event horizon and eventually reaches the singularity without being affected by any potential evaporation of the black hole.
It's worth noting that the concept of black hole evaporation, as described by Stephen Hawking's theory of Hawking radiation, applies to very small black holes over incredibly long timescales. For astrophysical black holes, the timescale for significant evaporation is far longer than the age of the universe, so it wouldn't cause the black hole to evaporate before infalling matter reaches the singularity.
However, it's important to mention that our current understanding of black holes is based on general relativity, which does not incorporate quantum effects. The complete description of black holes requires a theory that unifies general relativity and quantum mechanics, such as a theory of quantum gravity. Such a theory could potentially modify our understanding of black hole singularities and their interactions with infalling matter. Ongoing research aims to address these questions and provide a more complete picture of black hole physics.