The integral-lift conjecture for the top Chern–Dold character of IP-theory

Let $\mathrm{IP}^d$ be the degree-$d$ cohomology theory associated with the invertible-phase $\Omega$-spectrum, and let

$$\operatorname{ch}_{\mathrm{top}}\colon \mathrm{IP}^d\longrightarrow \mathrm{H}\mathbb{Q}^{d+2}$$

be the top Chern–Dold character. Integral-lift conjecture. The top Chern–Dold character has an integral lift for $d=3,4$, but has no such lift for $d\geq 6$. The conjecture is motivated by the paper's computations concerning maps from integral cohomology into the Anderson dual of the Thom spectrum; the cases $d=3,4$ and the obstruction for $d\geq 6$ are presented as the expected pattern.