On the boundedness of $k$-algebra homomorphisms between $k$-affinoid algebras

Let $k$ be a complete non-archimedean non-trivial valued field. In this paper, we investigate whether every $k$-algebra homomorphism between $k$-affinoid algebras is automatically bounded. We show that this property holds if and only if either $\mathrm{\mid }{k}^{\times }{\mathrm{\mid }}^{\mathbb{Q}}={\mathbb{R}}_{>0}$ holds, or $k$ has positive characteristic and is $F$-finite.