How negative can forward-limit coefficients be once gravity is included

For a weakly coupled 4D effective field theory of a massless scalar coupled to Einstein gravity, with $2\to 2$ amplitude $M(s,t) = 8\pi G s^2/(-t) + c_2 s^2/\Lambda^4 + ...$, determine the lowest value of $c_2$ consistent with unitarity, analyticity, crossing and a UV completion whose amplitude Reggeizes (grows as $s^{2 + \alpha' t}$) above a scale $M_*$. Under this Regge assumption proposals give $c_2 \ge -\operatorname{O}(\Lambda^4/(M_{\mathrm{pl}}^2 M_*^2))$; smeared impact-parameter sum rules in $D = 4$ instead give $c_2 \ge -\operatorname{O}(G \Lambda^4/M^2)\operatorname{log}(M b_{\mathrm{IR}})$, where $M$ is the EFT cutoff and $b_{\mathrm{IR}}$ an infrared impact-parameter cutoff. An answer is the optimal bound in $D = 4$ with its assumptions stated, saying which form is optimal, or an explicit consistent UV completion that violates it.

A derivation from stated axioms of the optimal lower bound on $c_{2}$, checked against explicit string-theory amplitudes.