Short answers, pulled from the story.
The problem of other minds is an epistemological puzzle about whether it is possible to know that other people have minds. Because we can only observe behavior and not inner experience directly, knowledge of other minds is always indirect. No degree of behavioral sophistication can definitively guarantee that thought is occurring within another person.
The problem of other minds is a direct extension of solipsism, the philosophical notion that for any person only one's own mind is known to exist with certainty. Solipsism raises the possibility that the inner lives of others are inferred rather than observed, making the existence of other minds permanently uncertain from a strictly logical standpoint.
In 1953, Karl Popper proposed that the test for the problem of other minds is whether one would seriously argue with the other person or machine. He reasoned that in arguing with others we unavoidably attribute intentions and mental states to them, and that we do not argue with a thermometer. Genuine argument, in Popper's view, already presupposes a mind on the other side.
Theory of mind is the ability to spontaneously infer the mental states of others. It is proposed as the reason the problem of other minds does not disrupt everyday social interactions, supported by innate mirror neurons, a dedicated theory of mind mechanism, or a tacit background theory. Growing evidence that behavior results from cognition, which requires a brain and often involves consciousness, further grounds this capacity.
Christian List's many-worlds theory of consciousness is a philosophical model designed to reconcile the subjective nature of consciousness without lapsing into solipsism. List argues that metaphysical theories of consciousness face a quadrilemma in which at least one of four positions must be false: first-person realism, non-solipsism, non-fragmentation, and one world. The many-worlds theory is his proposed resolution.
Egocentric presentism is a concept proposed by philosopher Caspar Hare as a weak form of solipsism. Under this view, other persons can be conscious, but their experiences are simply not present in the way that one's own current experience is. It is related to perspectival realism, which holds that things within perceptual awareness have a defining intrinsic property that exists absolutely rather than relative to anything else.