Conjecture on special values of zpread polynomials

Let $Z_n(x)$ be the $n$th zpread polynomial, let $n\geq 1$ be an integer, and define the golden ratio by

$$\varphi=\frac{1+\sqrt{5}}{2}.$$

Special-value conjecture. The following equivalences hold:

1. $Z_n(2)=2$ if and only if $n\equiv 1\pmod 2$.
2. $Z_n(3)=3$ if and only if $n\equiv 1,2\pmod 3$.
3. $Z_n(2+\varphi)=2+\varphi$ if and only if $n\equiv 1,4\pmod 5$.
4. $Z_n(2+\varphi)=3-\varphi$ if and only if $n\equiv 2,3\pmod 5$.

These assertions record empirical congruence patterns for fixed points and special values of the zpread polynomials on $[0,4]$. The supplied text describes them as empirical observations and gives no evidence that they have been proved or disproved.