Existence conjecture for smooth higher-degree plane-filling curves

Let $r\geq 2$ be an integer. For $k\in\mathbb{F}_q$, let $C_{k,r}\subset\mathbb{P}^2$ be the plane-filling curve defined by

$$x^r (x^q y-xy^q)+y^r (y^q z-yz^q)+(z^r+kx^r)(z^q x-zx^q)=0.$$

Existence conjecture for $C_{k,r}$. There exists an integer $m=m(r)$ such that, for every finite field $\mathbb{F}_q$ with $q>m$ and characteristic not dividing $r$, there is some $k\in\mathbb{F}_q$ for which $C_{k,r}$ is smooth. Computations found only four exceptional pairs in the tested ranges, but the assertion is unproved in general.