Is there a SHA-256 value which hashes to itself?
This question here gives the probability of such a value, called a "fixed point" existing as 63% (based on assuming a "random" property of SHA256):

Cryptographic hash where the hashing function input is similar to the result

There is at least one such "fixed point" of the compression function that the SHA256 algorithm uses. An example is here:

Fixed point of the SHA-256 compression function

As explained in that answer, "[e]asy to find fixed-points are a known property of this construction", therefore this isn't a problem.