Arithmetic completeness of the reflection calculus with nabla

Let $T$ be a theory, and let ${\mathrm{RC}^\nabla\hspace{-0.7pt}}$ be the reflection calculus with modalities $\Diamond_n$ and $\nabla_n$. An arithmetical interpretation $*$ in $\overline{\mathfrak{G}}_T$ maps strictly positive modal formulas to $\overline{\mathfrak{G}}_T$ as specified by $\top^*=1_T$, $(A\land B)^*=(A^*\land_T B^*)$, $(\Diamond_n A)^*=R_n(A^*)$, and $(\nabla_n A)^*=\mathit{\Pi}_{n+1}(A^*)$. Arithmetic completeness means that whenever $A^*\vdash_T B^*$ for every arithmetical interpretation $*$, then $A\vdash_{{\mathrm{RC}^\nabla\hspace{-0.7pt}}}B$. Arithmetic completeness conjecture. If $T$ is arithmetically sound, then ${\mathrm{RC}^\nabla\hspace{-0.7pt}}$ is arithmetically complete: the converse of the soundness theorem holds. This would characterize derivability in the reflection calculus exactly by validity under all arithmetical interpretations in $\overline{\mathfrak{G}}_T$.