  The tautological ring $R^*(M)$ of a smooth manifold $M$ is the ring of
characteristic classes generated by the Miller-Morita-Mumford classes, and is
often more accessible than the ring of all characteristic classes of smooth
$M$-bundles.
  In this paper, we show that the Krull dimension of the tautological ring
vanishes for almost all manifolds homotopy equivalent to $\mathbb{H} P^2$
through a combination of new methods in rational homotopy theory developed by
Alexander Berglund and the family signature theorem.
