Optimality conjecture for tree-suspension transfer functions

Let $1\le \ell<k$, let $F$ be a $k$-graph with $\pi_{\ell}(F)=\beta$, and set $s=k-\ell$. For $0\le t<s$, let $\mathcal T^{(k)}_{\ell,t}(F)$ denote the corresponding tree suspension, and let $Q_{s,t+1}$ and $Q_{s,t+1-\ell}$ be the transfer polynomials used in the construction. Let $\gamma$ be the unique solution of

$$Q_{s,t+1}(x)=Q_{s,t+1-\ell}(x)-(1-\beta)x^s.$$

Tree-suspension optimality conjecture. Then

$$\pi_{\ell}(\mathcal T^{(k)}_{\ell,t}(F))=Q_{s,t+1}(\gamma)$$

for every $0\le t<s$.

The conjecture asks whether the natural two-part tree-suspension construction is always optimal. The paper proves sharpness only in the cases $t=0$ and $t=k-\ell-1$ when either $\ell\ge k/2$ or $\ell=1$, so the general assertion remains open.