The comparison obtuse constant conjecture for Alexandrov spaces

Let $M$ be a compact Alexandrov space with curvature bounded below, and let $\widetilde{\rm ob}(M)$ be its comparison obtuse constant. Let $D(M)$ be the double of $M$. For a noncompact Alexandrov space $M$, let $\widetilde{\rm ob}_{\infty}(M)$ be its comparison obtuse constant from infinity, and let ${\rm ob}_{\infty}(M)$ be the corresponding obtuse constant from infinity.

Comparison obtuse constant conjecture. The conclusions of the relevant compact results would remain valid with $\widetilde{\rm ob}(M)$ in place of ${\rm ob}(M)$ when $M$ is compact. Likewise, the relevant noncompact theorem would remain valid with $\widetilde{\rm ob}_{\infty}(M)$ in place of ${\rm ob}_{\infty}(M)$ when $M$ is noncompact.

The conjecture proposes that the comparison versions of the obtuse constants suffice for the paper's compact and noncompact conclusions. The supplied text gives no resolution; the referenced theorem and corollary are not included in the candidate span, so their exact conclusions cannot be reproduced here.