In this paper, we provide a new family of trivariate proper quasi-copulas. As an application, we show that $W^{3}$ - the best-possible lower bound for the set of trivariate quasi-copulas (and copulas) - is the limit member of this family, showing how the mass of $W^3$ is distributed on the plane $x+y+z=2$ of $[0,1]^3$ in an easy manner, and providing the generalization of this result to $n$ dimensions.