Borel-Tits conjecture on abstract homomorphisms of algebraic groups

Let $\mathbf{G}$ and $\mathbf{G}'$ be algebraic groups defined over infinite fields $k$ and $k'$, respectively. Let $\mathbf{G}^+$ denote the subgroup of $\mathbf{G}(k)$ generated by the $k$-points of split smooth connected unipotent $k$-subgroups. Borel-Tits conjecture. If $\rho:\mathbf{G}(k)\to\mathbf{G}'(k')$ is an abstract homomorphism such that $\rho(\mathbf{G}^+)$ is Zariski-dense in $\mathbf{G}'(k')$, then there exist a commutative finite-dimensional $k'$-algebra $B$, a ring homomorphism $f:k\to B$, and a rational $k'$-morphism $\sigma:R_{B/k'}({}_B\mathbf{G})\to\mathbf{G}'$ such that

$$\rho\vert_{\mathbf{G}^+}=\sigma\circ r_{B/k'}\circ F,$$

where $F$ is induced by $f$ and $r_{B/k'}$ is the canonical restriction-of-scalars isomorphism. The source calls this a major open question concerning standard descriptions of abstract homomorphisms; no resolution is supplied.