Existence of faithful Riordan representations of Sn in finite characteristic for n ≥ 4

Ascertain whether faithful Riordan representations of the symmetric group Sn exist over fields of finite characteristic when n ≥ 4, i.e., determine the existence of injective homomorphisms Sn → R(K) with K a finite-characteristic field for n ≥ 4.

Background

Theorem 3 proves that for n ≥ 3, Sn cannot be embedded into the Riordan group over C, showing strong obstructions in characteristic zero. In contrast, Section 3 constructs faithful Riordan representations for S3 over Z3 and GF(3^q).

This raises the natural unresolved question: does Sn (n ≥ 4) admit faithful Riordan representations over some fields of finite characteristic, or do obstructions similar to the complex case persist?