Conjecture on largest minimum distances of binary self-orthogonal codes of dimension 5

Let $d(n,k)$ denote the largest minimum distance of a binary linear $[n,k]$ code, and let $d_{so}(n,k)$ denote the largest minimum distance of a binary self-orthogonal $[n,k]$ code. If

$$n=\sum_{i=0}^{k-1}\left\lceil \frac{d}{2^i} \right\rceil$$

for $d$ a single even integer, then the largest-minimum-distance conjecture.

$$d_{so}(n,k)=d(n,k)-2.$$

Otherwise,

$$d_{so}(n,k)=2\left\lfloor \frac{d(n,k)}{2} \right\rfloor.$$

This conjecture predicts the largest minimum distances of binary self-orthogonal codes from the corresponding unrestricted binary linear-code distances. The supplied text gives supporting results for several dimension-$5$ parameter pairs, but provides no resolution of the general claim.