Conjecture on singular points outside the upper level set for a unit-circle obstacle

Let $\Omega$ be a unit circle, and let the upper level set (ULS) be the set where an optimal body's height attains its maximum. Assume that the ULS has nonempty interior. A singular point is a ridge point if it is a point where the body's surface has the ridge structure described in the preceding analysis, and let $\theta$, $\varphi_1$, and $\varphi_2$ denote the associated angular parameters. Singular-point conjecture. If the ULS has nonempty interior, then every singular point outside the ULS is a ridge point and satisfies

$$\theta \ne 0,\qquad \varphi_1=-\varphi_2.$$

This conjecture describes the singular curves observed numerically on the lateral surface of an optimal body for a unit-circle obstacle. The supplied text does not state whether the claim has been proved or disproved.