Optimal constant in the implication from c-strong Schur to K-Schur

Determine the optimal function K(c) such that every real Banach space with the c-strong Schur property is K(c)-Schur; in particular, ascertain whether the known bound K(c)=2c+1 can be improved to 2c or c+1.

Background

The paper recalls known relationships between the c-strong Schur property and the c-Schur property in real Banach spaces. A general result shows that c-strong Schur implies (2c+1)-Schur. However, the authors point out that the sharpness of the constant 2c+1 is unknown, raising the problem of identifying the exact optimal constant in this implication.

References

However, it is not clear, whether the constant $2c+1$ is optimal or whether it can be replaced by $2c$ or $c+1$.

— Lipschitz-free spaces over uniformly discrete metric spaces are 3-Schur  (2604.01875 - Cúth et al., 2 Apr 2026) in Section 2 (On quantification of the Schur property), paragraph after the proposition on relationships between c-strong Schur and c-Schur