Asymptotic estimate of absolute projection constants

In this note we construct a sequence of real, k-dimensional symmetric spaces $ Y^k $ satisfying $ \text{lim} \ \underset{k}{ \text{inf} } \ \lambda_k^S // \sqrt{k} \ \geq \ \text{lim} \ \underset{k}{\text{inf}} \ \lambda(Y^k, l_1) // \sqrt{k} \ \gt \ \underset{w in [0,a_2]}{\text{max}} h(w) \ \gt \ 1//(2 - \sqrt{2//\pi})$, where $\lambda_k^S$ is defined by (4) and $h(w)=a_1^2\sqrt{2//\pi}+2a_1\sqrt{a_2^2-w^2}+w\sqrt{a_2^2 - w^2}$ with $a_1 = 1// (2-\sqrt{2//\pi}) $ and $a_2=1-a_1$. This improves the lower bound obtained in [3], Th. 5.3 by $\text{max}_{w \in [0,a_2]} h(w)$