About Füredi's conjecture

Let $t$ be a non-negative integer and \mbox{\cal P}=\{(A_i,B_i)\}_{1\leq i\leq m} be a set-pair family satisfying $\mathrm{\mid }{A}_i\cap B_i\mathrm{\mid }\le t$ for $1\le i\le m$. \mbox{\cal P} is called strong Bollobás $t$-system, if $\mathrm{\mid }{A}_i\cap B_j\mathrm{\mid }>t$ for all $1\le i\mathrm{\ne }j\le m$. Füredi conjectured the following nice generalization of Bollobás’ Theorem: Let $t$ be a non-negative integer. Let \mbox{\cal P}=\{(A_i,B_i)\}_{1\leq i\leq m} be a strong Bollobás $t$-system. Then $$\sum _{i=1}^m\frac{1}{\left(\genfrac{}{}{0px}{}{\mathrm{\mid }{A}_i\mathrm{\mid }+\mathrm{\mid }{B}_i\mathrm{\mid }-2t}{\mathrm{\mid }{A}_i\mathrm{\mid }-t}\right)}\le 1.$$ We confirmed the following special case of Füredi’s conjecture along with some more results of similar flavor. Let $t$ be a non-negative integer. Let \mbox{\cal P}=\{(A_i,B_i)\}_{1\leq i\leq m} denote a strong Bollobás $t$-system. Define $a_i:=\mathrm{\mid }{A}_i\mathrm{\mid }$ and $b_i:=\mathrm{\mid }{B}_i\mathrm{\mid }$ for each $i$. Assume that there exists a positive integer $N$ such that $a_i+b_i=N$ for each $i$. Then $$\sum _{i=1}^m\frac{1}{\left(\genfrac{}{}{0px}{}{a_i+b_i-2t}{a_i-t}\right)}\le 1.$$