The inflection-point conjecture for the $\alpha$-norm of a parametrized probability vector

Let $n\ge 3$, let $\alpha\in(0,1)\cup(1,+\infty)$, and let $p\in[0,\frac{1}{n}]$. Write $\text{\boldmath }v_n(p)$ for the probability vector under consideration and $H_{\text{\scriptsize \boldmath }v_n}(p)$ for its Shannon entropy. Let $\chi_n(\alpha)$ denote a possible inflection-point entropy.

Inflection-point conjecture. There exists an inflection point $\chi_n(\alpha)\in\bigl(H_{\text{\scriptsize \boldmath }v_n}(\mathrm{e}^{-n}),\ln n\bigr)$ such that $\|\text{\boldmath }v_n(p)\|_\alpha$ is strictly concave as a function of $H_{\text{\scriptsize \boldmath }v_n}(p)$ on $[0,\chi_n(\alpha)]$ and strictly convex on $[\chi_n(\alpha),\ln n]$. Moreover, $\chi_n(\alpha)$ is strictly increasing for $\alpha\in(0,1)\cup(1,\infty)$, and

$$\lim_{\alpha\to0^+}\chi_n(0)>H_{\text{\scriptsize \boldmath }v_n}(\mathrm{e}^{-n}),\qquad \lim_{\alpha\to1}\chi_n(\alpha)=\ln 2+\ln\sqrt{n-1},\qquad \lim_{\alpha\to+\infty}\chi_n(\alpha)=\ln n.$$

This is presented as a more accurate description of the preceding lemma's convexity and concavity behavior. The supplied text gives no evidence that the claim has been proved or disproved, so its status remains open.