I guess the following inequality

$$ \sum_{i=1}^n g \left (\frac{-p_i \log p_i}{H(\boldsymbol{p})} \right ) \le \sum_{i=1}^n g (p_i)$$

holds for any continuous convex function $g$ and any probability vector $\boldsymbol{p}=(p_1,\dots,p_n)\ge 0$ with $\sum_{i=1}^np_i=1$ where $H(\boldsymbol{p})=-\sum_{i=1}^n p_i\log p_i $ denotes the Shannon entropy of the probability distrbution $\boldsymbol{p}$. This is equivalent to that the following majorization relation holds

$$ \frac{-\boldsymbol{p}\log \boldsymbol{p}}{H(\boldsymbol{p})} \prec \boldsymbol{p}.$$

I already have a proof for the case where all probabilities are less than $e^{-1}$, but the general case where one or two of the probabilities can be larger than $e^{-1}$ has remained unsolved so far. The conjecture was numerically checked for $n=2, 3, 4$.

The problem may be of a particular theoretical interest because it seems not easy to obtain it using the existing procedures of generating majorization relations, as $-x\log x$ is not monotone.

The problem was first asked in this MSE question, and even after offering a bounty, I had no progress in approaching a full proof or constructing a counterexample (more details, related results and questions, and my past attempts can be found in that question).