  We study the spectrum of a family of algebras, the inhomogeneous Gaudin
algebras, acting on the $n$-fold tensor representation $\mathbb{C}[x_1, \ldots,
x_r]^{\otimes n}$ of the Lie algebra $\mathfrak{gl}_r$. We use the work of
Halacheva-Kamnitzer-Rybnikov-Weekes to demonstrate that the
Robinson-Schensted-Knuth correspondence describes the behaviour of the spectrum
as we move along special paths in the family. We apply the work of
Mukhin-Tarasov-Varchenko, which proves that the rational Calogero-Moser phase
space can be realised as a part of this spectrum, to relate this to behaviour
at $t=0$ of rational Cherednik algebras of $\mathfrak{S}_n$. As a result, we
confirm for symmetric groups a conjecture of Bonnaf\'e-Rouquier which proposes
an equality between the Calogero-Moser cells they defined and the well-known
Kazhdan-Lusztig cells.
