Serre's Conjecture I for smooth connected linear algebraic groups

Let $K$ be a perfect field whose cohomological dimension is less than or equal to $1$, and let $G$ be a smooth connected linear algebraic group over $K$.

Serre's Conjecture I. Does

$$H^1(K,G)=1$$

hold?

The conjecture was proved by Steinberg in 1965. Here $H^1(K,G)$ denotes the first Galois cohomology set, and the condition on the cohomological dimension is Serre's condition that the Brauer group vanishes over every algebraic extension of $K$.