Let $ k<N$ be natural numbers. In this question we consider graphs whose vertices are size-$ k$ subsets of a size-$ N$ universe. Consider the following random walk in the graph:

Starting from a set $ R$ pick $ t$ elements in $ R$ uniformly at random and pick a uniformly random set $ S$ that contains those $ t$ elements ($ t$ is a parameter; note that the size of the intersection of $ S$ and $ R$ may be larger than $ t$ ).

This model is studied in the association schemes literature and has an elegant spectral analysis (see, e.g., https://www.math.uwaterloo.ca/~cgodsil/pdfs/assoc2.pdf).

My question is whether one can prove a hypercontractive inequality in this model (equivalently, a Log Sobolev constant), similarly to what’s proven for closely related models in https://projecteuclid.org/download/pdf_1/euclid.aop/1022855885.