Zhao–He conjecture that joint set systems attain the optimal joints bound

For positive integers $k_1,\ldots,k_r,m_1,\ldots,m_r$, let $C_{k_1,\ldots,k_r;m_1,\ldots,m_r}$ be the optimal constant in the corresponding joints problem, and let $C^G_{k_1,\ldots,k_r;m_1,\ldots,m_r}$ be the optimal constant restricted to generically induced joint set systems. Zhao–He conjecture. For all such positive integers,

$$C^G_{k_1,\ldots,k_r;m_1,\ldots,m_r}=C_{k_1,\ldots,k_r;m_1,\ldots,m_r}.$$

This asserts that the set-theoretic generically induced model captures the exact optimal joints constant; the paper describes it as a weaker form of the broader conjecture and gives no resolution.