Optimality of the lower $d_1$–$J$ comparison constant

Let $(X,\omega)$ be a Kähler manifold of complex dimension $n$, and let $m$ be the constant in the lower bound comparing the $d_1$ distance to the $J$-functional on the level set $I(u)=0$. The conjectured value is

$$m=\frac{2}{n+1}\left(\frac{n}{n+1}\right)^n.$$

Optimality conjecture. The constant $m=\frac{2}{n+1}\left(\frac{n}{n+1}\right)^n$ is optimal for all Kähler manifolds $(X,\omega)$.

This value is known to be optimal in the toric setting, while its optimality for general Kähler manifolds remains open.