Possible complementation constants above two

Determine whether every complemented ideal has complementation constant exactly 2, or whether complemented ideals with complementation constants strictly larger than 2 can occur.

Background

For a proper ideal $\mathcal I$, the space $c_{0,\mathcal I}$ is a closed subspace of $\ell_\infty$, and the paper studies when it is complemented there. The complementation constant is the infimum of the norms of projections from $\ell_\infty$ onto $c_{0,\mathcal I}$. The paper proves a universal lower bound of 2 and establishes that the separable-injectivity constant is exactly 2 for every proper ideal.

The unresolved issue is whether the lower bound is always attained for complemented ideals, or whether some complemented ideals require projections of norm strictly greater than 2. The authors note that, by their projection formula, this is equivalent to asking whether the canonical quotient map admits a contractive linear right inverse whenever it admits a bounded one.