  The lower spectral radius of a set of $d \times d$ matrices is defined to be
the minimum possible exponential growth rate of long products of matrices drawn
from that set. When considered as a function of a finite set of matrices of
fixed cardinality it is known that the lower spectral radius can vary
discontinuously as a function of the matrix entries. In a previous article the
author and J. Bochi conjectured that when considered as a function on the set
of all pairs of $2 \times 2$ real matrices, the lower spectral radius is
discontinuous on a set of positive (eight-dimensional) Lebesgue measure, and
related this result to an earlier conjecture of Bochi and Fayad. In this
article we investigate the continuity of the lower spectral radius in a
simplified context in which one of the two matrices is assumed to be of rank
one. We show in particular that the set of discontinuities of the lower
spectral radius on the set of pairs of $2 \times 2$ real matrices has positive
seven-dimensional Lebesgue measure, and that among the pairs of matrices
studied, the finiteness property for the lower spectral radius is true on a set
of full Lebesgue measure but false on a residual set.
