The extremal-shade conjecture for cross-t-intersecting families

Let $C(n,k,l,t)$ be the collection of cross-$t$-intersecting pairs $(\mathcal A,\mathcal B)$, and define

$$N_0(n,m_k,m_l,k,l,t)=\max_{(\mathcal A,\mathcal B)\in C(n,k,l,t)}|\nabla_{m_k}(\mathcal A)|\,|\nabla_{m_l}(\mathcal B)|.$$

For $0\le i+j\le n-t$, define $\mathcal G_{ij}(n,k,t)=\{F\in\binom{[n]}{k}:|F\cap[t+i+j]|\ge t+i\}$. The cross-shade extremal-family conjecture. One has

$$N_0(n,m_k,m_l,k,l,t)=\max_{\substack{0\le i\le k-t\\0\le j\le l-t\i+j\le n-t}}|\mathcal G_{ij}(n,m_k,t)|\,|\mathcal G_{ji}(n,m_l,t)|.$$

The conjecture is the shade analogue of the proposed extremal description for cross-$t$-intersecting families. The paper gives the definitions and corresponding shade lemma but does not report a resolution.