Cotwist conjecture for functor categories on spheres

Let $\mathcal{D}$ be a stable $\infty$-category and let $n\teq 2$. Consider the cotwist functor $T_{\operatorname{Fun}(S^n,\mathcal{D})}$ of the spherical adjunction

$$f^*:\mathcal{D}\leftrightarrow \operatorname{Fun}(S^n,\mathcal{D}):f_*.$$

For an $\mathbb{E}_\infty$-ring spectrum $R$, set $\mathcal{D}=\operatorname{RMod}_R$, and let $\varphi:R[t_{n-1}]\rightarrow R[t_{n-1}]$ be the equivalence determined by

$$\varphi(t_{n-1})=(-1)^{n-1}t_{n-1}.$$

Cotwist conjecture. There is a commutative diagram in $\operatorname{LinCat}_R$ identifying $T_{\operatorname{Fun}(S^n,\operatorname{RMod}_R)}$ under the equivalence $\operatorname{Fun}(S^n,\operatorname{RMod}_R)\simeq\operatorname{RMod}_{R[t_{n-1}]}$ with $\varphi^*[-n]$. Moreover, if $S^n_{\operatorname{top}}$ is the topological $n$-sphere and $r:S^n_{\operatorname{top}}\rightarrow S^n_{\operatorname{top}}$ is the antipodal map, then

$$T_{\operatorname{Fun}(S^n,\mathcal{D})}\simeq r^*[-n].$$

This predicts that the cotwist is controlled by the antipodal involution together with a shift by $-n$, and gives an algebraic description for module categories over an $\mathbb{E}_\infty$-ring spectrum. The statement is presented as a conjecture because the preceding argument does not directly generalize to the spectral setting.