co.combinatorics - Reference request: The stable Kronecker ring is isomorphic to the ring of symmetric polynomials

For $lambda$ any partition and $n$ a positive integer, write $lambda[n]$ for the sequence $(n - |lambda|, lambda_1, lambda_2, ldots, lambda_r)$. For $n$ large enough, this is a partition of $n$.

The irreducible representations of $S_n$ are indexed by partitions of $n$; we denote them by $S_{lambda}$. The Kronecker coefficients $g_{lambda mu}^{nu}$ are defined by the equality
$$S_{lambda} otimes S_{mu} cong bigoplus g_{lambda mu}^{nu} S_{nu}$$
of $S_n$ representations.

It is a theorem of Murnaghan that $g_{lambda[n] mu[n]}^{nu[n]}$ becomes constant as $n to infty$. This constant value is called the stable Kronecker coefficient, and denoted $overline{g}_{lambda mu}^{nu}$. It is also a result of Murnaghan that, for given $lambda$ and $mu$, there are only finitely many $nu$ for which $overline{g}_{lambda mu}^{nu} neq 0$.

Therefore, we can define a commutative, associative ring to be spanned by the generators $overline{S}_{lambda}$, with relations
$$overline{S}_{lambda} overline{S}_{mu} = sum overline{g}_{lambda mu}^{nu} overline{S}_{nu}.$$

I'll call this the stable Kronecker ring.

I can prove that the stable Kronecker ring is isomorphic to the ring of symmetric functions. Is this fact already in the literature?