The MAX-SAT limiting function conjecture

Let $F_k(n,m)$ be a uniformly random $k$-SAT formula with $n$ variables and $m$ clauses, and let

$$f_k(n,m)=\mathbb{E}(\max F_k(n,m)),$$

where $\max F_k(n,m)$ denotes the maximum number of clauses simultaneously satisfied by an assignment. Fix $k$ and a constant $c>0$.

MAX-SAT limiting function conjecture. For every $k$ and every constant $c>0$, as $n\rightarrow\infty$,

$$\frac{f_k(n,cn)}{n}$$

converges to a limit.

The conjecture concerns the existence of an asymptotic expected optimum for random MAX-$k$-SAT. The source says it extends naturally to arbitrary CSPs and is open even for MAX-2-SAT; monotonicity in the clause density is known, but monotonicity in $n$ is not.