The Lawvere-theoretic conjecture for homotopy conilpotent groups

Let $n$ be a nonnegative integer, let $\mathcal{P}^n$ be a theory in the sense of Lawvere, and let its objects be the free homotopy conilpotent groups of class $n$, namely products of $\Sigma \mathrm{P}^n\mathbb{I}$ applied to products of $1$-spheres. A functor is $n$-co-excisive when it has the corresponding dual excision property. Lawvere-theoretic conjecture. For each $n$, there should be a weak equivalence of categories between the values taken by functors of the form $\Sigma F$, where $F$ is $n$-co-excisive, and homotopy conilpotent groups of class $n$. This would provide a dual Lawvere-theoretic description of the formally dual Goodwillie tower and clarify the relationship between co-excisive functors and homotopy conilpotent groups; the source does not establish the expected equivalence.