Universal scalable convergence conjecture for permuton-random sequences

A permuton is a probability measure on $[0,1]^2$ with uniform marginals; a sequence of permutations is $\Gamma$-random when it is sampled according to the permuton $\Gamma$, and it is scalably convergent when its pattern densities converge at every admissible scale.

Universal scalable convergence conjecture. If $\Gamma$ is any permuton, then every $\Gamma$-random sequence is scalably convergent.

The conjecture would extend the observed scalable convergence of sequences sampled from the permuton $\Gamma_\textsf{V}$ to every permuton. The supplied text presents this as a belief and gives no resolution.