Dimension-five self-orthogonal distance conjecture

Let $d(n,5)$ denote the largest minimum distance of a binary linear $[n,5]$ code, and let $\dso(n,5)$ denote the largest minimum distance of a binary self-orthogonal $[n,5]$ code. For integers $n\ge 10$ satisfying $n\neq 13$ and $n\equiv_{31}6,13,14,21,22,28,29$, consider optimal self-orthogonal codes of length $n$ and dimension $5$. Dimension-five self-orthogonal distance conjecture.

$$\dso(n,5)=d(n,5)-2.$$

Equivalently, there are no $[n,5,d(n,5)]$ self-orthogonal codes. The conjecture is based on the tabulated optimal self-orthogonal codes and computations using random codes for the indicated parameter range; the source gives no proof or resolution for all such $n$.