and Bethe ansatz diagonalizability) and which demonstrate universal asymptotic fluctuation behaviors. There are two main routes which have proved effective so far in producing integrable probabilistic systems — Macdonald processes (and various degenerations) [@BorodinCorwin2011Macdonald], [@BorodinGorinSPB12], [@BorodinPetrov2013Lect], and stochastic vertex models [@Borodin2014vertex], [@CorwinPetrov2015], [@BorodinPetrov2016_Hom_Lectures].

Prototypical examples within both classes of integrable models are the TASEP (Totally Asymmeric Simple Exclusion Process) [@Spitzer1970] and the PushTASEP, its counterpart with a long-range pushing mechanism [@liggett1980long], [@derrida1991dynamics]. In the context of *Macdonald processes*, both systems are connected to certain probability distributions on integer partitions $\lambda=(\lambda_1\ge\ldots\ge\lambda_N\ge0 )$ with nice algebraic structure, and the TASEP and PushTASEP are related to the distributions of, respectively, the smallest and the largest parts of $\lambda$. This picture can be generalized within the Macdonald process hierarchy to produce 2-parameter $q$-deformed discrete and continuous time TASEPs and PushTASEPs [@BorodinCorwin2013discrete], [@BorodinPetrov2013NN], [@CorwinPetrov2013], [@MatveevPetrov2014]. On the other hand, TASEP also belongs to the class of exactly solvable *stochastic vertex models*, and hence it admits a 3-parameter generalization to the $q$-Hahn TASEP [@Povolotsky2013], [@Corwin2014qmunu] (recalled in ). The latter is, in fact, a special case of the 4-parameter family of stochastic higher spin six vertex models studied in [@CorwinPetrov2015], [@BorodinPetrov2016_Hom_Lectures].

For some time it was 