A weighted Complete Intersection conjecture for Lagrangians

Let $G$ be a $t$-intersecting $r$-graph, meaning that any two edges of $G$ have intersection of size at least $t$. For integers $t\geq 1$, $i\geq 0$, and $r\geq t+i$, let $F(r,t,i)$ be the $(\leq r)$-graph on $[t+2i]$ whose edges have sizes between $t+i$ and $r$. For a weighting $p:[t+2i]^+\to[0,1]$, write $w_p(F(r,t,i))$ for the corresponding weighted edge sum, and require $(F(r,t,i),t+2i,p)$ to be a weighted $t$-intersecting set system. Weighted Complete Intersection conjecture. Then

$$r!\lambda(G)\leq\max_{0\leq i\leq r-t}w_p(F(r,t,i)),$$

where the maximum is implicitly taken over all such weightings $p$. This would extend the Complete Intersection Theorem from edge counts to Lagrangians of $t$-intersecting $r$-graphs; the supplied text presents it as a question toward a complete intersection theorem, and gives no resolution.