  We show that the multi-type stationary distribution of the totally asymmetric
simple exclusion process (TASEP) scales to a nontrivial limit around the
Bernoulli measure of density $1/2$. This is obtained by showing that the TASEP
speed process, introduced by Amir, Angel and Valk\'o, scales around the speed
$v=0$ to the stationary horizon (SH), a function-valued stochastic process
recently introduced and studied by the authors, SH is believed to be the
universal scaling limit of Busemann processes in the KPZ universality class.
Our results add to the evidence for this universality by connecting SH with
multiclass particle configurations. Previously SH has been associated with the
exponential corner growth model, Brownian last-passage percolation, and the
directed landscape.
