The plane radial-density conjecture for circles, undularies, and pinched circles

Consider the plane with perimeter density $r^{k}$, where $k>-1$, and area density $r^{m}$. The relevant candidate isoperimetric curves are circles about the origin, undularies, and pinched circles through the origin.

Plane radial-density conjecture. For $k\in[0,\infty)$, the following curves are isoperimetric:

$$\begin{array}{ll} m\in\left(k-1,k-1+\frac{1}{k+1}\right] &: \text{the circle about the origin},\\ m\in\left(k-1+\frac{1}{k+1},k-1+\frac{k+1}{2k+1}\right) &: \text{undularies},\\ m\in\left[k-1+\frac{k+1}{2k+1},k\right] &: \text{pinched circles through the origin}. \end{array}$$

This fills the area-density range between $k-1$ and $k$ omitted by the preceding proposition, by transferring the sector transition values through the power change of coordinates. The supplied text does not report a resolution.