The cycle/non-efficient contribution bound for stable primitivity rank

Let $\mathbb{\mathbf{F}}$ be a free group. Let $1\ne w\in\mathbb{\mathbf{F}}$ be a non-power, let $\sigma\in S_{d}$, and suppose that

$$\Gamma_{w^{\sigma}}\xrightarrow{\eta_{1}}\Sigma\xrightarrow{\eta_{2}}\Sigma'\xrightarrow{\eta_{3}}\Omega$$

is an element of ${\cal D}\mathrm{ecomp}_{\mathrm{alg}}^{3}(w^{\sigma}\to\mathbb{\mathbf{F}})$ such that $\eta_{1}$ is efficient, $\Sigma$ has a cycle as a connected component, $\Sigma'$ has none, and $\eta_{2}\circ\eta_{1}$ is not efficient. Cycle/non-efficient contribution conjecture. Under these hypotheses,

$$C_{\eta_{2}}^{\mathrm{alg}}(N)=O\left(N^{-d\mathrm{s\pi}(w)-1}\right).$$

This bound is the remaining technical estimate needed for the paper's asymptotic formulas for stable symmetric-group characters; proving it would show that the only leading contributions arise from the specified efficient diagrams.