On sizes of $1$-cross intersecting set pair systems

Let $\{(A_i,B_i)\}_{i=1}^m$ be a set pair system. Füredi, Gyárfás, and Király called it $1$-cross intersecting if $|A_i\cap B_j|$ is $1$ when $i\neq j$ and $0$ if $i=j$. They studied the systems and their generalizations and, in particular, considered $m(a,b,1)$, the maximum size of a $1$-cross intersecting set pair system in which $|A_i|\leq a$ and $|B_i|\leq b$ for all $i$. Füredi, Gyárfás, and Király proved that $m(n,n,1)\geq 5^{(n-1)/2}$ and asked whether there are upper bounds on $m(n,n,1)$ significantly better than the classical bound ${2n\choose n}$ of Bollobás for cross intersecting set pair systems. Answering one of their questions, Holzman proved recently that if $a,b\geq 2$, then $m(a,b,1)\leq \frac{29}{30}\binom{a+b}{a}$. He also conjectured that the factor $\frac{29}{30}$ in his bound can be replaced by $\frac{5}{6}$. Our goal is to prove this bound.