Cyclic subspace code existence conjecture for parameters n and k

Let $q$ be a prime power and let $k$ and $n$ be positive integers with $n\ge 2k$. A subspace code is a collection of $k$-dimensional subspaces of $\mathbb{F}_{q^n}$, and it is cyclic if it is closed under multiplication by every nonzero element of $\mathbb{F}_{q^n}$. The minimum subspace distance is the minimum of $\mathsf{d_S}(U,V)$ over distinct codewords $U,V$.

Cyclic subspace code existence conjecture. There exists a cyclic subspace code

$$\mathcal{C}\subseteq \mathcal{G}_q(n,k)$$

with minimum distance $2k-2$ and cardinality

$$\frac{q^n-1}{q-1}.$$

The conjecture asks for cyclic subspace codes attaining a full orbit of size $(q^n-1)/(q-1)$ while having intersection dimension at most $1$ between distinct codewords. The paper's Sidon-space constructions resolve an open question concerning the square span of a Sidon space, but the supplied context does not establish that this conjecture itself has been solved.