Determine the sharp barycenter-constrained Sobolev constant

Determine whether the infimum \(\mathfrak{a}_n\) for \(n\geq 5\), defined by the improved Sobolev quotient on \(S^n\) under the constraint \(\int_{S^n}x|u|^{2n/(n-2)}=0\), equals \(c_n^*=\frac{n(n-2)}{4}\bigl(2^{2/n}-1\bigr)\); equivalently, prove or disprove the conjectured identity \(\mathfrak{a}_n=c_n^*\).

Background

The second part of the paper studies the best improvement constant in the Aubin–Sobolev inequality on the sphere for functions whose critical-power barycenter vanishes. The authors define $\mathfrak{a}_n$ as the infimum of the corresponding normalized deficit and obtain a universal lower bound $\mathfrak{a}_n>\frac{n(n-2)}{4(n^2-3n+1)}$ for $n\geq5$.

A sequence of even functions with two concentration points gives the upper bound $$\mathfrak{a}_n\leq c_n^*=\frac{n(n-2)}{4}(2^{2/n}-1)$$. The exact optimal value is not established in the paper; the authors explicitly conjecture that this upper bound is attained. They also note that Dolbeault had conjectured the same equality in the class of even functions.