I'm trying to understand the X3DH protocol used in Signal and many other end to end encrypted messaging apps. There are other questions on this site asking about why the signed prekey exists. I'm asking why the prekey need to be signed.
In section 3.3 of the specification, 4 (3 if the one-time prekey is missing) Diffie-Hellman exchanges are done. DH1 involves the identity key of one user, and DH2 involves the identity key of the other user. The SK is derived from these exchanges in addition to the other exchanges that do not involve identity keys. Notably, DH4 is computed without involving any identity key or signed key.
How does any security property of X3DH change if the signature on the signed prekey is not validated?