Positivity of the quadratic overshoot coefficients in the NPA hierarchy

For each finite NPA level $k$, let $a_k$ be the coefficient in the small-$s$ expansion

$$c_k(s)=4-s+a_k s^2+O(s^3),$$

where the coefficients satisfy $a_k\downarrow0$ as $k$ increases. Positivity conjecture. $a_k>0$ for every finite $k$.

This would imply that every finite NPA level fails to be exact sufficiently close to the critical line, while the required exact level still remains finite for each fixed $s>0$ and diverges as $s\to0^+$. The claim is identified in the source as an open question of Gigena and is consistent with convergence of the NPA hierarchy.