Tieredness conjecture for scalable convergence of 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. A permuton is tiered if every vertical strip, after rescaling to a permuton, is identical to the original permuton.

Tieredness conjecture. If $\Gamma$ is a permuton for which every $\Gamma$-random sequence is scalably convergent, then $\Gamma$ is tiered.

Tiered permutons have this property because every rescaled vertical strip has the same pattern densities as the whole permuton. The conjecture asserts that these are the only permutons for which all $\Gamma$-random sequences are scalably convergent; the supplied text gives no resolution.