Classification conjecture for Sidon polynomials over prime fields

Let $p$ be a prime, and let $P\in\mathbb{F}_p[x]$ be a Sidon polynomial, meaning that $P$ parametrizes a Sidon set in the associated polynomial construction. For polynomials $P,P'\in\mathbb{F}_p[x]$, write $P\sim_{\mathcal S}P'$ when

$$P'=R\circ P\circ T$$

for some permutation polynomials $R,T\in\mathbb{F}_p[x]$.

Sidon-polynomial classification conjecture. If $P\in\mathbb{F}_p[x]$ is a Sidon polynomial, then either

$$P\sim_{\mathcal S}x$$

or

$$P\sim_{\mathcal S}x^2.$$

This is the reformulation of the maximum-Sidon-set classification in terms of Sidon-equivalence classes. The supplied source gives no evidence that the conjecture has been resolved.