Banach spaces with unconditional basis have w-FPP

Question

A Banach space $X$ is said to have w-FPP (weak fixed point property) if for every non-empty, weakly compact and convex subset $K\subseteq X$; every non-expansing mapping $T:K\longmapsto K$ i.e.

$$\|Tx-Ty\|\leq \|x-y\|\quad\quad \forall\, x,y \in K$$

has a fixed point.

Recall that if $X$ has an unconditional basis $(e_i)$, the constant of unconditionality $\lambda$ is

\begin{align} \lambda:=&\sup\lbrace \|\sum_{i=1}^{\infty} \epsilon_i\zeta_i e_i\| \;\; : \;\; \|\sum_{i=1}^{\infty} \zeta_i e_i \|=1\, , \, \epsilon_i=\pm 1\rbrace \\ =& \sup\lbrace \|2P_F-I\|\;\;:\;\; F\subset\mathbb{N}\rbrace \end{align}

where $P_F$ are the standard projections onto a finite subset $F$.

Suppose $X$ has constant of unconditionality $\lambda$. Lin¹ proved via ultraproduct constructions that if $\lambda<\frac{\sqrt{33}-3}{2}$ then $X$ has the w-FPP. This raises the question whether every Banach space with unconditional basis enjoys the w-FPP.

My questions are the following:

1. Has this result been improved in any direction? For instance, with further assumptions can the bound be improved? Do you know any references?
2. Are there any other approaches to tackle this problem?

Actually any reference or idea on how to proceed would help.

¹Lin, Pei-Kee. Unconditional bases and fixed points of nonexpansive mappings. Pacific J. Math. 116 (1985), no. 1, 69--76; MR769823, https://projecteuclid.org/euclid.pjm/1102707248, https://msp.org/pjm/1985/116-1/index.xhtml