Non-invariance of regularized Riesz energies for even-dimensional submanifolds

Let $M$ be a closed submanifold of Euclidean space, let $m=\dim M$, and let $E_M(z)$ denote its regularized Riesz energy. The scale-anomaly term $R_M(-2m)$ measures the failure of $E_M(-2m)$ to be scale invariant. Even-dimensional non-invariance conjecture. If $m$ is even, then $E_M(-2m)$ is not scale invariant; equivalently, there exists a submanifold $M$ such that

$$R_M(-2m)\ne0.$$

The paper proves Möbius invariance of $E_M(-2m)$ when $m$ is odd. This conjecture asserts contrasting behavior in even dimensions and would rule out scale invariance for at least one even-dimensional submanifold.