Reiher–Schacht conjecture on constant rooted clique densities

Let $r\ge 3$ and let $W$ be a graphon. For $x,y\in[0,1]$, write $t_{x,y}(K_r^{\bullet\bullet},W)$ for the rooted homomorphism density of the clique $K_r$ with two distinguished roots, and let $V_W^{(r)}(x,y)=t_{x,y}(K_r^{\bullet\bullet},W)$. Suppose that

$$V_W^{(r)}\equiv d$$

for some $d\in[0,1]$. Reiher–Schacht conjecture. Then $W$ is $K_r$-free when $d=0$, or

$$W\equiv d^{1/{\binom r 2}}.$$

This characterizes graphons whose rooted $K_r$-density graphon is constant: apart from the $K_r$-free case, the graphon should be constant. The conjecture is posed for $r\ge 3$ and is attributed in the source to concluding remarks of Reiher and Schacht; no resolution is given here.