Osipov's uniqueness conjecture for Sharygin triangles in regular polygons

Let a Sharygin triangle be a non-isosceles triangle whose bisectral triangle, formed by the intersections of its internal angle bisectors with the opposite sides, is isosceles. Suppose that the vertices of a Sharygin triangle coincide with three vertices of a regular polygon. Sharygin triangle uniqueness conjecture. Then this triangle is similar to the heptagonal triangle, whose angles are

$$\angle A=\pi/7,\quad \angle B=2\pi/7,\quad \angle C=4\pi/7.$$

The conjecture asserts that the heptagonal triangle is the only similarity type arising from three vertices of a regular polygon; the source abstract says that the paper proves this conjecture by a simple method.