Injectivity of the Atiyah–Patodi–Singer invariant for two-dimensional representations

Let $D_{2^{k+1}(2r+1)}$ be the indicated finite group, and let $\varrho_{t,s}$ denote its two-dimensional irreducible representations, with $1\leq t\leq 2r$ and $0\leq s\leq 2^{k-1}-1$. Injectivity conjecture. If $t_1\equiv t_2\mod 2$ and

$$\tilde{\xi}_{\varrho_{t_1,s}}(D)=\tilde{\xi}_{\varrho_{t_2,s}}(D),$$

then $t_1=t_2$. This is proposed as the remaining step needed to classify the two-dimensional irreducible representations by their vectors of Cheeger-Chern-Simons numbers; the source gives no resolution.