Let $R$ be a complete DVR (or just Henselian) with field of fractions $F$ and algebraically closed residue field $k$ (of characteristic $p>0$). Let $G$ be a reductive group scheme over $R$ and let $T$ be a maximal torus of the generic fibre $G_F=G\times_{R} F$.

Let $\overline{T}$ be the schematic closure of $T$ in $G$. Explicitly, if $T=\mathrm{Spec}\,F[x_1,\dots,x_n]/I$, then $$ \overline{T}=\mathrm{Spec}\,R[x_1,\dots,x_n]/(IF[x_1,\dots,x_n]\cap R[x_1,\dots,x_n]). $$ Then $\overline{T}$ is a flat group scheme over $R$ with generic fibre $T$.

Assume that $k$ has very good characteristic for $G$. Is $\overline{T}$ smooth over $R$? In other words, does $\overline{T}$ have smooth/reduced special fibre?

For $G=\mathrm{GL}_{n}$ this is true because $T$ is then the multiplicative group of an algebra. I think I can also prove this for $G=\mathrm{SL}_{n}$, but the argument does not generalise.

A weaker property than smoothness which is still enough for my purposes, is the following:

Is the canonical map $\rho:\overline{T}(R)\rightarrow\overline{T}(k)$ surjective?