The converse to chromatic blueshift for finite abelian p-groups

Let $p$ be a prime, let $A$ be a finite abelian $p$-group, and let $\mathcal{F}$ be a proper family of subgroups of $A$. Let $R$ be a commutative algebra regarded as a Borel-equivariant genuine $A$-spectrum with trivial action. Write $\operatorname{cork}_p(\mathcal{F})$ for the $p$-rank codimension associated to $\mathcal{F}$, and let $T(n)$ denote the height-$n$ telescopic localization theory. The converse to chromatic blueshift. If $R$ is $T(n)$-acyclic, then $\Phi^{\mathcal{F}}R$ is $T(n-\operatorname{cork}_p(\mathcal{F}))$-acyclic. The authors state that this is plausible but do not know a proof, even when $A=C_p$; it is proposed as a converse to the preceding blueshift theorem.