Uniform sphere-retraction conjecture for polyhedra of fixed dimension

Let $X$ be a polyhedron of dimension $d$, and let $B$ be a wedge of spheres. Write $\operatorname{tors}(X)$ and $\operatorname{tors}(B)$ for the torsion subgroups of the relevant stable homotopy groups, and let $1_X$ and $1_B$ denote the identity maps. Congruence modulo torsion is denoted by $\equiv\pmod{\operatorname{tors}}$.

Uniform sphere-retraction conjecture. For every integer $d$ there is an integer $m>0$ such that, for every polyhedron $X$ of dimension $d$, there are maps

$$X\xrightarrow{\beta}B\xrightarrow{\alpha}X$$

with

$$\alpha\beta\equiv m1_X\pmod{\operatorname{tors}(X)},\qquad \beta\alpha\equiv m1_B\pmod{\operatorname{tors}(B)}.$$

Theorem 34 shows that such a property would imply the bounded genus conjecture above. The paper presents this as a sufficient condition rather than proving it; its uniform assertion for each fixed dimension remains open in the supplied text.