Cohomological comparison conjecture for loop spaces of spin groups

Let $p$ be an odd prime, let $q$ be a power of $p$, and let $\mathbb{F}_q$ be the finite field with $q$ elements. For $n\geq 3$, let $\mathrm{Spin}(n)$ be the spin group, let $\bar{A}_{n-1}=p_n^{-1}(A_{n-1})$ be the specified subgroup of $\mathrm{Spin}(n)$, and let $\tilde{B}\bar{A}_{n-1}$ and the maps

$$\tilde{\alpha}_n:\tilde{B}\bar{A}_{n-1}\to\mathcal{L}B\mathrm{Spin}_n, \qquad \tilde{\alpha}_{n,q}:\tilde{B}\bar{A}_{n-1}\to\mathcal{L}_{\phi^q}B\mathrm{Spin}_n$$

be as defined in the source. The spin-group comparison conjecture. The induced homomorphisms

$$H^{*}(\mathcal{L}B\mathrm{Spin}_n)\stackrel{\tilde{\alpha}_n^*}{\longrightarrow}H^{*}(\tilde{B}\bar{A}_{n-1})\stackrel{\tilde{\alpha}_{n,q}^*}{\longleftarrow}H^{*}(\mathcal{L}_{\phi^q}B\mathrm{Spin}_n)$$

are monomorphisms, and the image of $\tilde{\alpha}_{n,q}^*$ is the same as the image of $\tilde{\alpha}_n^*$. This would give a common cohomological description of the ordinary and Frobenius-twisted loop spaces through the subgroup associated with $A_{n-1}$; the supplied source gives no resolution of the conjecture.