Essential tightness conjecture for the complete-graph rigidity parameter

Let $a_d(K_n)$ denote the parameter associated with $d$-dimensional algebraic connectivity for the complete graph $K_n$. Complete-graph tightness conjecture. If $n\geq 2d$, then

$$\left\lfloor\frac{n}{2d}\right\rfloor\leq a_d(K_n)\leq \frac{n}{2d}.$$

The conjecture asserts that the lower bound from the preceding theorem is essentially tight. The upper bound is included in the conjectured estimate, and no resolution is supplied in the source.