Algebraic connectivity bound for complete multipartite graphs

Let $d\geq 1$ and $k\geq 2$. For positive integers $n_1,\ldots,n_k$, write $n=n_1+\cdots+n_k$, and let $K_{n_1,\ldots,n_k}$ be the complete $k$-partite graph whose sides have sizes $n_1,\ldots,n_k$. The quantity $a_d(G)$ denotes the $d$-dimensional algebraic connectivity of a graph $G$. The complete multipartite connectivity conjecture. There exist constants $c_{d,k},M_{d,k}>0$ such that, for all $n_1,\ldots,n_k\geq M_{d,k}$,

$$a_d(K_{n_1,n_2,\ldots,n_k})\geq c_{d,k}\cdot\min\\{n-n_1,n-n_2,\ldots,n-n_k\\}.$$

For $d=1$ this is an equality, while for $d\geq 2$ the corresponding rigidity and exact-value questions remain open, particularly for complete multipartite graphs with at least three parts.