Modified nonexistence conjecture for optimal self-orthogonal codes of dimension 5

Let $d(n,5)$ denote the maximum minimum distance of a binary linear $[n,5]$ code, and let $d_{\mathrm{so}}(n,5)$ denote the maximum minimum distance of a binary self-orthogonal $[n,5]$ code. Modified nonexistence conjecture. If $n=14,21,22,28,29$, or if $n\ge 32$ and $n\equiv 6,13,21,28 \pmod{31}$, then

$$d_{\mathrm{so}}(n,5)=d(n,5)-2,$$

i.e., there are no $[n,5,d(n,5)]$ self-orthogonal codes. This is the complementary modified conjecture to the preceding existence claim, separating residue classes where optimal self-orthogonal codes are conjectured not to exist; the supplied text does not establish its resolution.