Let $F$ be a field and $A$ be an $F$-central simple algebra of degree $n$. Let $0< k< n$ and let $SB_k(A)$ denote the generalized Severi-Brauer variety: if $E/F$ is a field extension, $SB_k(A)(E)$ consists of the right ideals of dimension $kn$ of $A_E=A\otimes_F E$.
If $A$ is split, i.e. $A\simeq M_n(F)$, then $SB_k(A)=Gr(k,n)$, the Grassmannian.
Is the converse true? If not, can you provide a counterexample?
The result is true if $k=1$, since $SB_k(A)$ has a rational point over $F$ iff the index of $A$ divides $k$, and Grassmannians have rational points over $F$.
I've crossposted the question to MO: https://mathoverflow.net/questions/153150/when-are-generalized-severi-brauer-varieties-grassmannians