  The cut-rank of a set $X$ in a graph $G$ is the rank of the $X\times
(V(G)-X)$ submatrix of the adjacency matrix over the binary field. A split is a
partition of the vertex set into two sets $(X,Y)$ such that the cut-rank of $X$
is less than $2$ and both $X$ and $Y$ have at least two vertices. A graph is
prime (with respect to the split decomposition) if it is connected and has no
splits. A graph $G$ is $k^{+\ell}$-rank-connected if for every set $X$ of
vertices with the cut-rank less than $k$, $\lvert X\rvert$ or $\lvert
V(G)-X\rvert $ is less than $k+\ell$. We prove that every prime
$3^{+2}$-rank-connected graph $G$ with at least $10$ vertices has a prime
$3^{+3}$-rank-connected pivot-minor $H$ such that $\lvert V(H)\rvert =\lvert
V(G)\rvert -1$. As a corollary, we show that every excluded pivot-minor for the
class of graphs of rank-width at most $k$ has at most $(3.5 \cdot 6^{k}-1)/5$
vertices for $k\ge 2$. We also show that the excluded pivot-minors for the
class of graphs of rank-width at most $2$ have at most $16$ vertices.
