Existence of optimal cyclic subspace codes

Let $q$ be a prime power and let $k$ and $n$ be positive integers satisfying $n>2k$. A cyclic subspace code is a single-orbit code under scalar multiplication, and $\mathcal{G}_q(n,k)$ denotes the Grassmannian of $k$-dimensional subspaces of $\mathbb{F}_q^n$.

Existence conjecture. There exists a cyclic subspace code $\mathcal{C}\subseteq \mathcal{G}_q(n,k)$ such that

$$|\mathcal{C}|=\frac{q^n-1}{q-1}$$

and

$$d(\mathcal{C})=2k-2.$$

Such codes would attain the stated orbit size and minimum distance simultaneously for every prime power $q$ and all positive integers $k,n$ with $n>2k$.