Dual endomorphism conjecture for Anderson self-dual ring spectra

Let $R$ be an $\mathbf{E}_1$-ring spectrum and write $A=\pi_0 R$. Suppose that there is a class $D\in \pi_{-d}R$ witnessing the Anderson self-duality of $R$. Let $F\colon R\to R$ be an endomorphism of algebra objects in $\mathrm{h}\mathrm{Sp}$ such that

$$F(D)=\lambda D$$

for some $\lambda\in A$. Dual endomorphism conjecture. The composites $F\circ \widecheck{F}$ and $\widecheck{F}\circ F$ should be equivalent to multiplication by $\lambda$ on $\pi_\ast R$.

This conjecture describes how an endomorphism of an Anderson self-dual ring spectrum interacts with its dual endomorphism. The source presents it as a conjecture, and no resolution is given in the supplied text.