On the constant of Lipschitz approximability

In this note we find $\lambda >1$ and give an explicit construction of a separable Banach space $X$ such that there is no $\lambda$-Lipschitz retraction from $X$ onto any compact convex subset of $X$ whose closed linear span is $X$. This is closely related to a well-known open problem raised by Godefroy and Ozawa in 2014 and represent the first known example with such a property.