Pointwise Christoffel function estimate for two-dimensional $[?]$-balls

Let $1<\alpha<2$ and let ${\boldsymbol{x}}\in B^2_\alpha$. Define

$$\delta:={\rm dist}({\boldsymbol{x}},\partial B^2_\alpha),\qquad l_i:=\max\{t:{\boldsymbol{x}}+(-1)^it{\boldsymbol{v}}\in B^2_\alpha\},\quad i=1,2,$$

where ${\boldsymbol{u}}$ satisfies ${\boldsymbol{x}}+\delta{\boldsymbol{u}}\in\partial D$ and ${\boldsymbol{v}}$ is one of the two unit vectors orthogonal to ${\boldsymbol{u}}$. Pointwise Christoffel function conjecture. If $\delta>\sigma n^{-2}$, with $\sigma>0$, then

$$\lambda_n(B^2_\alpha,{\boldsymbol{x}})\approx c(\alpha,\sigma)n^{-2}\sqrt{l_1l_2}.$$

The preceding discussion establishes analogous estimates at singular and smooth boundary points, while the general pointwise behavior away from the diagonal direction is posed as an interesting problem; the source does not provide a resolution of this assertion.