Nonsingularity and odd-order initial conditions for weighted polyharmonic minimizers

Let $m$ and $\alpha$ be the parameters defining the weighted Sobolev quotient $\mathcal{S}(m,2,\alpha,\alpha,\infty)$, and suppose that this quotient admits a $0$-th nonsingular minimizer at $r=0$. Nonsingularity conjecture. The $0$-th nonsingular minimizers for $\mathcal{S}(m,2,\alpha,\alpha,\infty)$ are necessarily $(2m-1)$-th nonsingular at $r=0$ and satisfy the odd-order left-zero initial conditions. This assertion is known when $m=2$, while it is proposed in order to compute the best constant for $m>2$ under the stated existence assumption.