Random permutations from $q$-Demazure products

We study the $q$-deformation of the Demazure product model from arXiv:2407.21653. Consider the longest element $w_0$ in $S_n$ written as a reduced word in simple transpositions. Independently delete each transposition with probability $1-p$ and apply the $q$-Demazure product to the remaining ones. We show that the law of the resulting permutation converges as $n \to \infty$ to a deterministic permuton, which coincides with the $q=0$ case studied in arXiv:2407.21653 for adjusted probability $p'=p(1-q)/(1-qp)$. This resolves Conjecture 1.13 from arXiv:2407.21653 and identifies the limiting permuton explicitly.