Asymptotic equivariant coercivity conjecture for harmonic map lumps

For the $n$-lump sector of the ${\mathbb{C}}{{P}}^1$ model on $S^2$, let $\tau^{eq}(\mu)$ be the equivariant coercivity constant for the map $\phi:z\mapsto \mu z^n$, with $\mu\in[1,\infty)$. Asymptotic equivariant coercivity conjecture. For all $n\geq 1$, the limit

$$\lim_{\mu\rightarrow\infty}\tau^{eq}(\mu)$$

exists and is finite. For all $n>1$, this limit is not zero. The paper proves continuity of $\tau^{eq}(\mu)$ but states that it cannot yet rigorously establish the existence or value of the limit; the asserted nonzero limit for $n>1$ therefore remains open.