The conjectures in the title were formulated in the late 1970s as vast generalizations of the classical theorem of Stickelberger. They make a very subtle connection between the $\Bbb Z[G(F/k)]$--module structure of the Quillen K-groups K${_\ast}(O_F)$ in an abelian extension $F/k$ of number fields and the values at negative integers of the associated $G(F/k)$--equivariant $L$--functions $\Theta_{F/k}(s)$. These conjectures are known to hold true if the base field $k$ is $\Bbb Q$, due to work of Coates-Sinnott and Kurihara. In this talk, we will provide evidence in support of these conjectures over arbitrary totally real number fields $k$.