Let $X$ be an absolutely irreducible surface over an algebraically closed or finite field $k$ of characteristic $p$. Is it true that the following conditions are equivalent?

1. There is a purely inseparable $k$-rational dominant map $\varphi\!: X \to \mathbb{P}^2$.

2. There is a purely inseparable $k$-rational dominant map $\psi\!: \mathbb{P}^2 \to X$, i.e., $X$ is a generalized Zariski surface.

Are degrees of $\varphi$ and $\psi$ equal?

Let $X$ be an absolutely irreducible surface over an algebraically closed or finite field $k$ of characteristic $p$. Is it true that the following conditions are equivalent?

1. There is a purely inseparable $k$-rational dominant map $\varphi\!: X \dashrightarrow \mathbb{P}^2$.

2. There is a purely inseparable $k$-rational dominant map $\psi\!: \mathbb{P}^2 \dashrightarrow X$, i.e., $X$ is a generalized Zariski surface.

Are degrees of $\varphi$ and $\psi$ equal?