Kra–Moreira–Richter–Robertson logarithmic growth conjecture for finite sumsets

For $cdelta cin (0,1)$ and $Ncinmathbb{N}$, let $cphi_{cdelta}(N)$ be the largest integer such that every set $Acsubseteq [N]$ with $|A|cgeq cdelta N$ contains a sumset $B+C$, where $B,Ccsubseteq [N]$ and cminc{|B|,|C|cgeq cphi_{cdelta}(N). Kra–Moreira–Richter–Robertson conjecture. For every $cdeltacin(0,1)$,

$$0<climinf_{Nctocinfty}\frac{cphi_{cdelta}(N)}{clog(N)}$$

and

$$climsup_{Nctocinfty}\frac{cphi_{cdelta}(N)}{clog(N)}<cinfty.$$

This conjecture asserts that the largest guaranteed summands in dense finite subsets of $[N]$ have logarithmic order, bounded above and below by positive constants depending on the density. It was posed by Kra, Moreira, Richter and Robertson; its resolution is not established in the supplied source.