The deficiency-one conjecture for oriented Grassmannians

Let $\widetilde{\operatorname{Gr}}_k(n)$ denote the oriented Grassmannian, and let $\delta(\widetilde{\operatorname{Gr}}_k(n))$ denote its deficiency. The previously established cases are excluded. Deficiency-one conjecture. In all remaining cases not covered by the stated proposition, one has

$$\delta(\widetilde{\operatorname{Gr}}_k(n))=1$$

more explicitly, this holds for $5\leq k\leq 2^t-5$, except when $k$ is odd and $n=2^t$. The conjecture would, for $n\neq 2^t$, follow from the characteristic rank conjecture, while the exceptional odd case reflects an additional relation and is not covered.