Symmetric hyperbolic 4-cobordism conjecture

Let $M$ be a hyperbolic $4$-cobordism with boundary components $N_0$ and $N_1$. Suppose that $N_0$ and $N_1$ are highly symmetric to each other, both topologically and geometrically. Symmetric hyperbolic 4-cobordism conjecture. Then $(M;N_0,N_1)$ should be an $h$-cobordism, possibly nontrivial; that is, $M$ need not be homeomorphic to the product $N_0\times[0,1]$. The preceding examples show that hyperbolic homology cobordisms can be homotopically nontrivial when their boundary components differ substantially, while the conjecture predicts that strong topological and geometric symmetry between the boundary components forces the stronger $h$-cobordism property without necessarily forcing a product cobordism.