Let $P$ denote a parabolic subgroup scheme of $\text{Sp}(2n;F)$, where $F$ is a field (I am interested in $K=\mathbb{Q}_p$ so possibly okay to assume local with characteristic $0$ if it makes life easier).

Let $X$ be a smooth quasi-projective scheme over $F$, and let $T\longrightarrow X$ denote a principal $P$-bundle over $X$, i.e. a $P$-torsor over $X$. By definition, $P$ is etale locally trivial.

Question: Is $P$ also Zariski locally trivial?

A reference or a proof would be appreciated!

Let $P$ denote a parabolic subgroup scheme of $\operatorname{Sp}(2n;F)$, where $F$ is a field (I am interested in $K=\mathbb{Q}_p$ so possibly okay to assume local with characteristic $0$ if it makes life easier).

Let $X$ be a smooth quasi-projective scheme over $F$, and let $T\longrightarrow X$ denote a principal $P$-bundle over $X$, i.e. a $P$-torsor over $X$. By definition, $P$ is étale locally trivial.

Question: Is $P$ also Zariski locally trivial?

A reference or a proof would be appreciated!

Let $P$ denote a parabolic subgroup scheme of $\text{Sp}(2n;F)$, where $F$ is a field (I am interested in $K=\mathbb{Q}_p$ so possibly okay to assume local with characteristic $0$ if it makes life easier).

Let $X$ be a smooth quasi-projective scheme over $F$, and let $T\longrightarrow X$ denote a principal $P$-bundle over $X$, i.e. a $P$-torsor over $X$. Clearly, $P$ is etale locally trivial.

Question: Is $P$ also Zariski locally trivial?

A reference or a proof would be appreciated!

Let $P$ denote a parabolic subgroup scheme of $\text{Sp}(2n;F)$, where $F$ is a field (I am interested in $K=\mathbb{Q}_p$ so possibly okay to assume local with characteristic $0$ if it makes life easier).

Let $X$ be a smooth quasi-projective scheme over $F$, and let $T\longrightarrow X$ denote a principal $P$-bundle over $X$, i.e. a $P$-torsor over $X$. By definition, $P$ is etale locally trivial.

Question: Is $P$ also Zariski locally trivial?

A reference or a proof would be appreciated!