Limit-point conjecture for the top eigenvalue vectors of regular graphs

Let $d\geq 3$ and $k$ be fixed. For a $d$-regular graph $G$, write $\lambda_1(G)\geq\lambda_2(G)\geq\cdots$ for its ordered eigenvalues. Define $B(d,k)$ by

$$B(d,k)=\{(\mu_1,\mu_2,\ldots,\mu_k):d=\mu_1\geq\mu_2\geq\mu_3\geq\cdots\geq\mu_k\geq 2\sqrt{d-1}\}.$$

Top-eigenvalue vector limit-point conjecture. For any $d\geq 3$ and any fixed $k$, the set of all limit points of the vectors

$$(\lambda_1(G_i),\lambda_2(G_i),\ldots,\lambda_k(G_i))$$

along infinite sequences $G_i$ of $d$-regular graphs is exactly $B(d,k)$. This extends the proved one-coordinate result for the second eigenvalue, while the realizability of all admissible vectors remains open.