$1$-cross intersecting set pair systems are small

A set pair system $\{(A_i,B_i){\}}_{i=1}^m$ is $1$-cross intersecting if $\mathrm{\mid }{A}_i\cap B_i\mathrm{\mid }=0$ for all $i$ and $\mathrm{\mid }{A}_i\cap B_j\mathrm{\mid }=1$ whenever $i\mathrm{\ne }j$. Let $m(a,b,1)$ denote the maximum size $m$ of a $1$-cross intersecting set pair system $\{(A_i,B_i){\}}_{i=1}^m$ where $\mathrm{\mid }{A}_i\mathrm{\mid }\le a$ and $\mathrm{\mid }{B}_i\mathrm{\mid }\le b$ for all $i$. We prove a conjecture of Füredi, Gyárfás, and Király [Combin. Probab. Comput. 32 (2023)] that asserts $m(n,n,1)\mathrm{/}\left(\genfrac{}{}{0px}{}{2n}{n}\right)\to 0$ as $n\to \mathrm{\infty }$.