Gerbner et al.'s hemi-bundled Bollobás conjecture for intersecting families

Let $AK(n,k,t)$ denote the maximum size of a $k$-uniform $t$-intersecting family $F\subseteq {[n]\choose k}$. Let $\{(A_i,B_i)\}_{i=1}^m$ be a collection of pairs of sets such that, for every $i\in[m]$, $|A_i|=a\leq|B_i|=b$. Suppose that, for some $t\geq 0$,

$$|A_i\cap A_j|\geq t\quad\text{for all }1\leq i,j\leq m,$$ $$|A_i\cap B_i|=0\quad\text{for all }1\leq i\leq m,$$ $$|A_i\cap B_j|>0\quad\text{for all }1\leq i\neq j\leq m.$$

Gerbner et al.'s conjecture. Then

$$m\leq AK(a+b,a,t).$$

The conjecture is presented as a further research problem after the paper settles another conjecture of Gerbner et al. Its resolution is not stated in the supplied text.