  In this paper, we provide a new proof of the stable Adams conjecture. Our
proof constructs a canonical null-homotopy of the stable J-homomorphism
composed with a virtual Adams operation, by applying the $\mathrm{K}$-theory
functor to a multi-natural transformation. We also point out that the original
proof of the stable Adams conjecture is incorrect and present a correction.
This correction is crucial to our main application. We settle the question on
the height of higher associative structures on the mod $p^k$ Moore spectrum
$\mathrm{M}_p(k)$ at odd primes. More precisely, for any odd prime $p$, we show
that $\mathrm{M}_p(k)$ admits a Thomified $\mathbb{A}_n$-structure if and only
if $n < p^k$. We also prove a weaker result for $p=2$.
