Vô Conjecture 23(VC23)

We study equations of the form H(f(x))=H(g(x)),x\in\mathbb{N}^\ast,\bigmwhere f,gare integer-valued polynomials on the positive integers and His one of the classical multiplicative arithmetic functions: the Dedekind psi function \psi, Euler's totient function \varphi, or the sum-of-divisors function \sigma. Motivated by extensive examples and known results for linear shifts, we formulate and discuss a conjecture (Vô Conjecture 23) asserting that such equations admit infinitely many solutions if and only if both polynomials are linear. We provide heuristic justification, supporting examples, partial results, and possible strategies toward a proof.