MacDonald's conjecture on closest projections to nilpotent operators

Let $\mathcal{N}(\mathbb{C}^n)$ denote the set of nilpotent operators on $\mathbb{C}^n$, and let $\delta_n$ be the distance from the set of non-zero projections in $\mathbb{M}_n(\mathbb{C})$ to $\mathcal{N}(\mathbb{C}^n)$. Let $\nu_{1,n}$ denote the distance from the set of rank-one projections in $\mathbb{M}_n(\mathbb{C})$ to $\mathcal{N}(\mathbb{C}^n)$. MacDonald's conjecture states that the closest non-zero projections to $\mathcal{N}(\mathbb{C}^n)$ have rank one. MacDonald's conjecture.

$$\delta_n=\nu_{1,n}=\frac{1}{2}\sec\left(\frac{\pi}{n+2}\right)\quad\text{for all}\quad n\in\mathbb{N}.$$

The formula for $\nu_{1,n}$ was proved by MacDonald, and the conjectured equality is known for $n=3$ and has since been verified for $n=4$. It remains open for all $n\geq 5$.