Height-function limit for intermediate q-reduced pipe-dream permutations

For fixed $p,q\in(0,1)$, let $\boldsymbol{w}^{(q)}(n)\in S_n$ be the random permutation from the $q$-reduced pipe-dream model, and let $H^{(q)}$ be its height function. Let $\mathsf{h}^{(q)}$ be a height function on $[0,1]^2$. Height-function limit conjecture. The permutations $\boldsymbol{w}^{(q)}(n)$ converge to a permuton determined by $\mathsf{h}^{(q)}$, namely

$$\lim_{n\to\infty}n^{-1}H^{(q)}(\lfloor n\mathsf{x}\rfloor,\lfloor n\mathsf{y}\rfloor)=\mathsf{h}^{(q)}(\mathsf{x},\mathsf{y}),\qquad (\mathsf{x},\mathsf{y})\in[0,1]^2,$$

in probability. This is a proposed limit-shape statement for the intermediate $q$ regime; the source provides no resolution.