In most uses of Sponge mode of operations such as SHA3 and many of the round-2 candidates in the NIST lightweight cryptography project, the underlaying primitive is a cryptographic permutation - that is, it's bijective.
For reasons of curiosity, I noticed that a proposed Sponge-based PRNG implement forward secrecy (back-tracing resistance) by truncating part of the permutation state to all-bits-zero (Section 4.3 of the linked paper).
This got me wonder, what if the permutation is one-way (e.g. $p(x) \oplus x$):
Q1: What are the security implications if SHA3 hash and XOF functions were built on top of Sponge-PRF?
Q2: If the proposed PRNG used PRF, how would the security proof break apart?
Q3: If sponge-based duplex AEADs used PRF, how would their security proof break apart?