Let $S$ be a sub-ring of a commutative ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S\setminus \{0\} \implies a,b \in S$.

Let $k$ be an algebraically closed field of positive characteristic. Let $A$ be a factorially closed, finitely generated, $k$-sub-algebra of $k[X_1,X_2,X_3]$ such that $tr.deg_k A=2$. Then is it true that $A$ is a polynomial ring in $2$-variables over $k$ i.e. $A$ is isomorphic as a $k$-algebra to $k[T_1,T_2]$ ?

When $k$ is algebraically closed and of characteristic zero, then the answer is affirmative [Theorem 1, Factorially closed subrings of commutative rings, pp.1140]