I'm studying the Weil Restriction $\mbox{Res}_{k/\textbf{F}_p}$ where $\textbf{F}_p$ is the field with $p$ elements and $k$ is a perfect field over $\textbf{F}_p$, not necessarily finite.

In this case, it is known that the Weil restrictions for affine schemes over $k$ are representable by ind-schemes, and I want to know what happens for projective schemes.

More concretely, the question is: Is $\mbox{Res}_{k/\textbf{F}_p} \textbf{P}^1_k$ representable by an ind-scheme?

There are two things I have thought so far:

I tried to construct $\mbox{Res}_{k/\textbf{F}_p} \textbf{P}^1$ by trying to "glue" two copies of $\mbox{Res}_{k/\textbf{F}_p} \textbf{A}^1$ along the induced map $f: \mbox{Res}_{k/\textbf{F}_p} \textbf{G}_m \rightarrow \mbox{Res}_{k/\textbf{F}_p} \textbf{A}^1$, but it does not work since the map $f$ does not seem to be an "open immersion" (being invertible is not an open condition in the ind-scheme $\mbox{Res}_{k/\textbf{F}_p} \textbf{A}^1$).

Another way is to study the quotient of $\mbox{Res}_{k/\textbf{F}_p} \textbf{A}^2 - \{0\}$ by the action of $\mbox{Res}_{k/\textbf{F}_p} \textbf{G}_m$, but I do not know how to proceed.

Any help or suggestion would be greatly appreciated.