The excess bound conjecture for pairs of d-polytopes

Let $d\geq 4$ be an integer, and let $\mathcal{E}^d$ be the set of $(f_0,f_1)$-pairs of $d$-polytopes. For a pair $v=(p,q)$ with

$$p\geq d+1,\qquad \frac{d}{2}p\leq q\leq \binom{p}{2},$$

its excess is $2q-dp$.

Excess bound conjecture. If $v\notin\mathcal{E}^d$, then

$$2q-dp\leq 4d-10.$$

Equivalently, the set of $(f_0,f_1)$-pairs for $d$-polytopes with excess strictly larger than $3d-10$ is

$$\mathcal{E}_{>3d-10}^d=\left\{(f_0,f_1):\frac{d}{2}f_0+\frac{3d-10}{2}<f_1\leq\binom{f_0}{2}\right\}.$$

This conjecture proposes a uniform characterization of sufficiently high-excess $(f_0,f_1)$-pairs for all dimensions $d\geq 4$, extending the preceding dimension-specific results. Its resolution would determine exactly which pairs above the threshold $3d-10$ arise from $d$-polytopes; no general proof or counterexample is supplied here.