Can we find a universal constant $c>0$ such that for all $p,q\in\Delta:=\lbrace x\in (0,1)^{n}\ \colon\ x_{1}+\dots+x_{n}=1\rbrace$ it is true that
\begin{equation}
|p_{i}-q_{i}|\le c\left|\ln\frac{p_{i}}{g(p)}-\ln\frac{q_{i}}{g(q)}\right|,\quad i=1,\dots,n
\end{equation}
where $g(p):=\left(\prod_{i=1}^{n}p_{i}\right)^{\frac{1}{n}}$ is the geometric mean of $p$? The question is of interest since it upper bounds the Euclidean distance on $\Delta$ with the Aitchison distance used in compositional data analysis.

At least in dimension $n=2$ the answer is positive. Indeed, in that case the claim follows simply by the mean-value theorem:

\begin{align*}
\frac{|p_{1}-q_{1}|}{|\ln\frac{p_{1}}{g(p)}-\ln\frac{q_{1}}{g(q)}|}=\frac{2|p_{1}-q_{1}|}{|\ln\frac{p_{1}}{1-p_{1}}-\ln\frac{q_{1}}{1-q_{1}}|}=2\xi(1-\xi)\le1,
\end{align*}
where $\xi\in \overline{p_{1}q_{1}}$.

Unfortunately, so far a failed to prove the general case (if true at all) and I'm grateful for any suggestions. Thanks in advance.