In 2003, Gijsbers, Wales, and I formulated conjectures about the structure of the BMW algebras of type D_n (n > 3), and E_n (n = 6, 7, 8). Subsequently, we proved these conjectures for D_n (n > 3). Recently, Wales and I worked on E_n leading to proofs of freeness and cellularity and to a verification of the conjectured dimensions of the BMW algebras of the title. In this talk I will describe the algebras, the related Brauer and Temperley-Lieb algebras, the connections with root systems. If time allows, I will attempt a sketch of proof of the announced results.