The off-diagonal restricted online Ramsey number conjecture

Let $r(3,n)$ be the classical Ramsey number, and let $\tilde{r}(3,n;N)$ denote the restricted online Ramsey number on $N$ vertices. Off-diagonal restricted online Ramsey conjecture. There exists an absolute constant $c$ such that if $N=r(3,n)$, then

$$\tilde{r}(3,n;N) \le (1-c)\binom{N}{2}.$$

The off-diagonal case may allow larger savings than the diagonal case; the conjecture would follow if Builder either quickly obtains a blue $n$-clique or forces a constant fraction of the edges of $K_N$ to be blue.