Extension of the random Turán upper-bound theorems to $(c,r)$-bounded graphs

Let $F$ be a bipartite graph that is $(c,r)$-bounded with respect to a triple $(S,T,T^*)$, meaning that $S\cup T$ is a bipartition of $F$, $T^*\subseteq T$ consists of $c$ vertices adjacent to every vertex of $S$, and every vertex in $T\setminus T^*$ has degree at most $r$. Let $a(F)$ and $b(F)$ be the quantities defined in the source from the corresponding function $f$ for $(c,r)$-bounded graphs.

Extension conjecture. Theorems

$$and$$

should hold for $(c,r)$-bounded graphs with $1\le c\le r$.

The conjecture proposes extending the paper's principal technical results beyond the case $c=1$, which the authors note recovers their previous definitions. The required analogues of the preliminary lemmas are asserted to be obtainable after modifications, but the generalized theorems are not proved in the supplied text.