The multi-degree bound conjecture for higher Betti numbers of realizable sign conditions

Let $\mathcal{P}$ and $\mathcal{Q}$ be finite sets of polynomials in $\mathbb{R}[X_1,\ldots,X_k]$, with the same notation and hypotheses as in Theorem B-B; in particular, let $s=|\mathcal{P}|$, let $d_1$ and $d_2$ denote the relevant degree bounds, and let $k'$ be the parameter appearing there. For a sign condition $\sigma\in\{0,1,-1\}^{\mathcal{P}}$, write $\mathrm{Reali}(\sigma,\mathrm{Zer}(\mathcal{Q},\mathbb{R}^k))$ for its realization on the common zero set of $\mathcal{Q}$, and let $b_i(-,\mathbb{Z}_2)$ denote the $i$-th Betti number with coefficients in $\mathbb{Z}_2$. The multi-degree bound conjecture. For all $i$ with $0\leq i\leq k'$, one has

$$\sum_{\sigma\in\{0,1,-1\}^{\mathcal{P}}} b_i\bigl(\mathrm{Reali}(\sigma,\mathrm{Zer}(\mathcal{Q},\mathbb{R}^k)),\mathbb{Z}_2\bigr)\leq O(1)^k s^{k'-i}d_1^{k-k'}d_2^{k'}.$$

This conjecture asks whether the known multi-degree bound for the sum of connected components of realizable sign conditions extends to all Betti numbers. Such higher-Betti-number bounds would be useful in incidence questions in discrete geometry; the question was previously raised in work of Barone and Basu.