Unique indecomposable basis conjecture for linear subspace codes

Let $\mathcal{U}$ be a linear code in the projective space $\mathbb{P}_q(n)$. An indecomposable basis is a basis of the vector space over $\mathbb{F}_2$ formed by the code whose elements are indecomposable codewords. The code $\mathcal{U}$ is closed under intersection when $U\cap V\in\mathcal{U}$ for all $U,V\in\mathcal{U}$. Unique indecomposable basis conjecture. A linear code $\mathcal{U}$ in $\mathbb{P}_q(n)$ has a unique indecomposable basis if and only if $\mathcal{U}$ is closed under intersection.

The conjecture concerns the relationship between the internal basis structure of a linear subspace code and closure under the meet operation of the projective lattice. The source presents it as an open question, motivated by examples of codes with more indecomposable codewords than their dimension.