Austin's common fixed-point conjecture for commuting contractions

Let $(X,d)$ be a complete metric space, let $r$ be a positive integer, and let $\lambda\in(0,1)$. A map $f:X\to X$ is a $\lambda$-contraction if

$$d(f(x),f(y))\leq\lambda d(x,y)$$

for all $x,y\in X$. The maps $f_1,\ldots,f_r$ pairwise commute if $f_i\circ f_j=f_j\circ f_i$ for all $i,j$.

Austin's conjecture. If $\{f_1,\ldots,f_r\}$ is a pairwise commuting family of $\lambda$-contractions on $X$, then the maps have a common fixed point; that is, there exists $x\in X$ such that $f_i(x)=x$ for every $i$.

The source reports that Austin proved the case $r=2$, while Milićević proved the case $r=3$ and the case of sufficiently small $\lambda$; the general statement remains open.