Statistics of irreducible components in large tensor powers of the spinor representation for $\mathfrak{so}_{2n+1}$ as $n\to\infty$

We consider the Plancherel measure on irreducible components of tensor powers of the spinor representation of $\mathfrak{so}_{2n+1}$. With respect to this measure, the probability of an irreducible representation is the product of its multiplicity and dimension, divided by the total dimension of the tensor product. We study the limit shape of the highest weight as the tensor power $N$ and the rank $n$ of the algebra tend to infinity with $N/n$ fixed.