Nonexistence conjecture for extremizers of the odd fractional Schrödinger inequality

Let $p\geq 2$, let ${\bf Q}_p$ denote the sharp constant in the odd convolution inequality, and let ${\bf C}_2$ be the best constant for the parabola in convolution form, with

$${\bf C}_2^6=\frac{\pi}{\sqrt{3}}.$$

Conjecture for the odd fractional Schrödinger inequality. For every $p\geq 2$,

$$\biggl(\frac{{\bf Q}_p}{{\bf C}_2}\biggr)^6=\frac{5}{p(p-1)}.$$

Moreover, extremizers for the odd sharp convolution inequality do not exist.

For $1<p<2$, the preceding discussion establishes existence of extremizers, while the case $p\geq 2$ remains harder; the conjecture asserts that the critical threshold is attained as a sharp constant but not by any extremizer.