Curvature-dimension conjecture for the α-Grushin plane

Let $b1 geqslant 1$. The $b1$-Grushin plane is denoted by $b1$-Grushin plane, and $b1$ and $N$ are the parameters of the synthetic curvature-dimension condition $\mathrm{MCP}(K,N)$. Let $m\binom{}{}$ be the unique nonzero solution in $b1nterval{-3}{-2}$ of

$$(m+1)^{2 \alpha}(m+1)-\left((2\alpha+1)m+1\right)=0.$$

Curvature-dimension conjecture. For $b1 geqslant 1$, the $b1$-Grushin plane satisfies $\mathrm{MCP}(K,N)$ if and only if $K\leqslant 0$ and

$$N\geqslant 2\left[\dfrac{(\alpha+1)m+1}{m+1}\right].$$

The conjecture proposes the sharp synthetic curvature-dimension range for the $b1$-Grushin plane, extending the study of measure contraction properties in sub-Riemannian spaces. Its resolution is not supplied in the source.