The BKK-bound growth conjecture for second-order averaged functions

Let $H_2(n,m)$ denote the quantity of second-order averaged functions under consideration, and let $u_{n,m}$ be its BKK bound. BKK-bound growth conjecture.

$$\nu_{n,m}=\nu_{3,m}\times(2m-1)^{n-3}.$$

This conjecture is motivated by the numerical BKK bounds in the paper, including the recently proved value $H_2(4,2)=9$. Its general validity is not established in the supplied text.