The mixed random fewnomial square-root bound conjecture

For $i=1,\ldots,n$, let $A_i\subseteq\mathbb{R}^n$ be finite nonempty sets of cardinality $t_i$, with convex hulls $P_i$. Let $V_0$ be the number of vertices of the Minkowski sum $P_1+\cdots+P_n$. Write $\mathbb{E}(A_1,\ldots,A_n)$ for the expected number of positive zeros of the corresponding random system. Mixed random fewnomial square-root bound conjecture. There is a function $\kappa:\mathbb{N}^2\to\mathbb{N}$ such that

$$\mathbb{E}(A_1,\ldots,A_n)\leq\kappa(n,V_0)\sqrt{t_1\cdots t_n}.$$

In particular, for $A\subseteq\mathbb{R}^n$ of cardinality $t$, one has

$$\mathbb{E}(A,\ldots,A)\leq\kappa(n,V_0)t^{\frac n2}.$$

The product-support construction preceding the conjecture gives a lower bound of order $\sqrt{t_1\cdots t_n}$, so the claim asserts that this order is optimal up to a factor depending only on $n$ and the number of vertices of the Minkowski sum. Its resolution status is not established by the supplied text.