  Stembridge introduced the notion of stability for Kronecker triples which
generalize Murnaghan's classical stability result for Kronecker coefficients.
Sam and Snowden proved a conjecture of Stembridge concerning stable Kronecker
triple, and they also showed an analogous result for Littlewood--Richardson
coefficients. Heisenberg coefficients are Schur structure constants of the
Heisenberg product which generalize both Littlewood--Richardson coefficients
and Kronecker coefficients. We show that any stable triple for Kronecker
coefficients or Littlewood--Richardson coefficients also stabilizes Heisenberg
coefficients, and we classify the triples stabilizing Heisenberg coefficients.
We also follow Vallejo's idea of using matrix additivity to generate Heisenberg
stable triples.
