Divisibility conjecture for twisted characteristic numbers of the Hopkins–Mahowald manifold

Let $M_1$ be the $24$-dimensional String manifold of Hopkins–Mahowald. For non-negative integers $i$, $j$, and $k$, let $T^i\otimes\wedge^j\otimes S^k$ denote the corresponding tensor product of the tangent-bundle, exterior-power, and symmetric-power constructions. Divisibility conjecture. For any non-negative integers $i$, $j$, and $k$,

$$24\mid \widehat{A}(M_1,T^i\otimes\wedge^j\otimes S^k),$$

and

$$24\mid \operatorname{Sig}(M_1,T^i\otimes\wedge^j\otimes S^k).$$

The claim extends the computer verification reported in the source for $i+j+k\leq 5$ and predicts universal divisibility by $24$ for these twisted $[?]$-genus and signature quantities. Its general validity is not established in the supplied text.