Stable compressed-rank conjecture for Stallings graphs

Let $B$) be a basis and let $\Gamma$ be a Stallings graph. For $d\in\mathbb{N}$, write $s\overline{\pi}_d(\Gamma)$ for the stable compressed rank and let $\overline{\pi}(\Gamma)$ denote the corresponding compressed rank. Let $\overline{\operatorname{Crit}}(\Gamma)$ be the set of connected $B$-core graphs $\Delta$ such that

$$-\chi(\Delta)=\overline{\pi}(\Gamma)-1$$

and there is a morphism $\Gamma\to\Delta$. A $d$-cover of $\Gamma$ inside $\Gamma\times_{\Omega_B}\Delta$ and a $f(\Delta')$-covering have their usual graph-theoretic meanings.

Stable compressed-rank conjecture. For every Stallings graph $\Gamma$ and $d\in\mathbb{N}$,

$$s\overline{\pi}_d(\Gamma)=\overline{\pi}(\Gamma)-1.$$

Moreover, if a Stallings graph $\Delta$ has a $d$-cover of $\Gamma$ inside $\Gamma\times_{\Omega_B}\Delta$ and $-\chi(\Delta)=s\overline{\pi}_d(\Gamma)$, then there is a function $f\colon\overline{\operatorname{Crit}}(\Gamma)\to\mathbb{Z}_{\geq 1}$ with

$$\sum_{\Delta'\in\overline{\operatorname{Crit}}(\Gamma)}f(\Delta')=d$$

and $\Delta$ is the disjoint union, over $\Delta'\in\overline{\operatorname{Crit}}(\Gamma)$, of an $f(\Delta')$-covering of $\Delta'$.

The stable compressed rank is not known to be integral or independent of $d$; the conjecture predicts both properties and describes all extremal cases as coverings of the critical graphs.