Embedding conjecture for optimal dimension-four binary codes

Let $[n,4]$ codes be binary linear codes of length $n$ and dimension $4$. A code is optimal if its minimum distance is $d(n,4)$, and it is self-orthogonal if it is orthogonal to itself. An optimal code is embedded to an optimal self-orthogonal code if it can be extended to a self-orthogonal code that is optimal among self-orthogonal codes, with maximum minimum distance $\dso(n,4)$. Embedding conjecture. Any optimal $[n,4]$ code can be embedded to an optimal self-orthogonal code. This conjecture is motivated by computations for optimal codes of dimensions $2$, $3$, and $4$ and lengths up to $256$, together with the examples in the paper; the source gives no general proof or resolution.