The Whitehead asphericity conjecture. Let $X$ be a $2$-dimensional aspherical simplicial complex and let $Y \subset X$ be a connected subcomplex. The conjecture then is that $Y$ is aspherical.
Very little is known about this, but a deep theorem of Bestvina and Brady says that the Eilenberg-Ganea conjecture and the Whitehead asphericity conjecture cannot both be true.
