Asymptotic growth of shuffle squares over large alphabets

For $k\in\mathbb{N}$, let $[k]=\{1,\ldots,k\}$ and let $\mathcal{S}_k(n)$ be the set of words in $[k]^{2n}$ that can be partitioned into two disjoint, identical subwords. A word in $\mathcal{S}_k(n)$ is a shuffle square, and $n$ is its semi-length. Since $|\mathcal{S}_k(n)|$ is supermultiplicative in $n$, define

$$b_k=\lim_{n\to\infty}|\mathcal{S}_k(n)|^{1/(2n)}.$$

Large-alphabet shuffle-square growth conjecture. As a function of $k\in\mathbb{N}$,

$$\lim_{n\to\infty}|\mathcal{S}_k(n)|^{1/n}=4k-2-o(1).$$

The conjecture proposes that the known upper bound is essentially tight for large fixed alphabets. The binary case is solved by the paper's main theorem, while the corresponding behavior for larger alphabets remains open; in particular, the paper separately asks whether $|\mathcal{S}_3(n)|=(9-o(1))^n$.