The universality conjecture for $T_{2n-1}(p^r)$

Let $p$ be a prime, let $n,r$ be positive integers, and let $P^{2n}(p^r)\xrightarrow{i}T_{2n-1}(p^r)$ be the canonical map. A space $Z$ is strongly $(2n,r)$-subexponential if

$$p^{r+k}\pi_i(Z)=0$$

for all $i\leqslant 2np^{k+1}$ and $k\geqslant 0$. An Abelian H-space is an H-space whose multiplication is homotopy commutative and homotopy associative. Universality conjecture. The map

$$P^{2n}(p^r)\xrightarrow{i}T_{2n-1}(p^r)$$

is universal for targets which are strongly $(2n,r)$-subexponential Abelian H-spaces: every map from $P^{2n}(p^r)$ to such a target extends, in the relevant H-space sense, through $T_{2n-1}(p^r)$. This proposed strengthening is intended to make $T_m(p^r)$ an acceptable target for the program of constructing universal Abelian H-spaces; the supplied text gives no resolution, so the conjecture remains open.