The double smoothness conjecture for Alexandrov spaces with maximal obtuse constant

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

Double smoothness conjecture. If $M$ is compact and ${\rm ob}(M)=\pi/2$, then $D(M)$ has no singular points. Similarly, if $M$ is noncompact and ${\rm ob}_{\infty}(M)=\pi/2$, then $D(M)$ has no singular points.

This conjecture predicts that the extremal value of the obtuse constant forces the double of an Alexandrov space to be nonsingular. The supplied text gives no resolution or further evidence beyond the conjectural statement.