  Let $M$ be an isoparametric hypersurface in the sphere $S^n$ with four
distinct principal curvatures. M\"{u}nzner showed that the four principal
curvatures can have at most two distinct multiplicities $m_1, m_2$, and Stolz
showed that the pair $(m_1,m_2)$ must either be $(2,2)$, $(4,5)$, or be equal
to the multiplicities of an isoparametric hypersurface of FKM-type, constructed
by Ferus, Karcher and M\"{u}nzner from orthogonal representations of Clifford
algebras. In this paper, we prove that if the multiplicities satisfy $m_2 \geq
3m_1 - 1$, then the isoparametric hypersurface $M$ must be of FKM-type.
Together with known results of Takagi for the case $m_1 = 1$, and Ozeki and
Takeuchi for $m_1 = 2$, this handles all possible pairs of multiplicities
except for 10 cases, for which the classification problem remains open. The
paper improves the result of a pre-existing preprint with the same title, in
which 14 cases remained open.
