The asymptotic vanishing conjecture for cross-intersecting shades

Let $N_1(n,m,k,t)=N_0(n,m,m,k,k,t)$, where $N_0$ is the maximum product of the corresponding shades of a pair of cross-$t$-intersecting families. Assume

$$k(m)=o(m),\qquad \lim_{m\to\infty}k(m)=\infty,\qquad \lim_{m\to\infty}\frac{t(m)}{\sqrt{k(m)}}=\infty.$$

The cross-shade asymptotic conjecture. Then

$$\lim_{m\to\infty}\frac{\sqrt{N_1(2m,m,k(m),t(m))}}{\binom{2m}{m}}=0.$$

This is the cross-intersecting analogue of the preceding asymptotic conjecture, concerning the normalized geometric mean of the two shade sizes. The source presents it as the final asymptotic conjecture and gives no resolution.