The positivity conjecture for derivatives of the sum of squared logarithms

Let $e=(e_1,\ldots,e_n)\in\mathbb{R}_+^n$, and let $f\circ\varphi$ be the composition defined in the paper, where $\varphi$ parametrizes the roots associated with $e$. For $k\in\{1,\ldots,n-1\}$, consider the partial derivative $\frac{\partial(f\circ\varphi)}{\partial e_k}(e)$. Positivity conjecture. The composition $f\circ\varphi$ is differentiable, and

$$\frac{\partial(f\circ\varphi)}{\partial e_k}(e)>0$$

for all $e\in\mathbb{R}_+^n$ and every $k\in\{1,\ldots,n-1\}$. This is the monotonicity condition needed in the paper's proof of the sum of squared logarithms inequality; the supplied material does not establish whether the assertion is resolved.