The 4-torsion conjecture for oriented Grassmannians

Let $\widetilde{\operatorname{Gr}}_k(n)$ be the oriented Grassmannian, and let $a_n$ and $d_n$ denote the anomalous generators used in the source. 4-torsion conjecture. The following assertions hold:

1. If $k\leq 4$, all torsion in $\operatorname{H}^*(\widetilde{\operatorname{Gr}}_k(n);\mathbb{Z})$ is $2$-torsion.
2. If $1<k<2^t-1$, $k$ is odd, then all torsion in $\operatorname{H}^*(\widetilde{\operatorname{Gr}}_k(2^t);\mathbb{Z})$ is $2$-torsion.
3. If $k\geq 6$ is even, $k\leq n-k$, and

$$c=\min(\deg a_n,\deg d_n)-1=\min(k(n-2^{t-1})+2^{t-1}-2,2^t-2),$$

then $\operatorname{H}^{c+1}(\widetilde{\operatorname{Gr}}_k(n);\mathbb{Z})$ contains a $4$-torsion class. 4. If $k\geq 5$ is odd, $5\leq k\leq 2^{t-1}<n<2^t$, $t\geq 5$, and

$$2^t-1=\deg d_n<\deg a_n=k(n-2^{t-1})+2^{t-1}-1,$$

equivalently $n>\frac{k+1}{k}2^{t-1}$, then $\operatorname{H}^{2^t-1}(\widetilde{\operatorname{Gr}}_k(n);\mathbb{Z})$ contains a $4$-torsion class.

The conjecture combines known absence results for 4-torsion with predicted infinite families where 4-torsion occurs. The source presents it as an expectation supported by the preceding proposition and experimental evidence; no resolution is supplied.