Bounded genus conjecture for polyhedra of fixed dimension

Let $X$ be a finite cell complex, or polyhedron, in the stable homotopy category. Two polyhedra are in the same genus when their localizations at every prime are stably equivalent. Let $g(X)$ denote the number of stable isomorphism classes in the genus of $X$.

Bounded genus conjecture. For every positive integer $n$ there is an integer $c_n$ such that

$$g(X)\le c_n$$

for every polyhedron $X$ of dimension $n$.

This is proposed as an analogue of Roiter's boundedness theorem for genera of modules over an order in a semisimple algebra. The paper does not establish the conjecture, so boundedness uniformly over all polyhedra of a fixed dimension remains open.