The super-exponential gap conjecture for off-diagonal Ramsey numbers

Let $R(\ell,k)$ be the two-colour Ramsey number, and suppose that $k$ and $\ell$ are sufficiently large with $\log k\ll\ell\ll k$. Super-exponential gap conjecture. For every fixed $C>0$,

$$R(\ell,k)\leqslant e^{-C\ell}\binom{k+\ell}{\ell}.$$

The conjecture says that the then-current bound is still super-exponentially far from the truth in the regime $\log k\ll\ell\ll k$; the source gives no resolution.