Mahowald's conjecture on higher associativity of cofiber spectra

Let $\tau$ be a nonzero element of $\pi_{k-1}(\mathbb{S}^0)$, and let $C\tau$ denote its cofiber. Mahowald's conjecture. The spectrum $C\tau$ does not admit an $A_{\infty}$-structure. This generalizes the expected failure of infinite homotopy associativity for Moore spectra and remains open in the stated generality.