Let $n>1$, and let $f:\mathbb{R}^n_{\ge 0}\rightarrow\mathbb{R}_{\ge 0}$ be continuously differentiable, concave, and homogeneous of degree one. Here, homogeneity of degree one means that for all $s\in\mathbb{R}_{\ge 0}$, and $x\in\mathbb{R}^n_{\ge 0}$, $f(sx)=sf(x)$.
And suppose that there exist continuously differentiable monotonic increasing functions $g:\mathbb{R}\rightarrow\mathbb{R}_{\ge 0}$ and $h_1,\dots,h_n:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R}$ such that for all $x\in\mathbb{R}^n_{\ge 0}$: $$f(x)=g(h_1(x_1)+\cdots+h_n(x_n)).$$
Must it be the case that there exists $a_1,\dots,a_n\in\mathbb{R}$ and $\rho,b_1,\dots,b_n\in\mathbb{R}_{\ge 0}$ such that for all $i\in\{1,\dots,n\}$, $h_i(x)=a_i+b_i \frac{x^{1-\rho}-1}{1-\rho}$? (Where, when $\rho=1$, we understand this as stating $h_i(x)=a_i+b_i\log(x)$.)
Note that by Euler's homogeneous function theorem: $$g(h_1(x_1)+\cdots+h_n(x_n))=(h_1'(x_1)x_1+\cdots+h_n'(x_n)x_n)g'(h_1(x_1)+\cdots+h_n(x_n)).$$
Does this (differential equation) help?
I cross posted this on economics.se as I thought there might be more chance of an answer there (at https://economics.stackexchange.com/questions/54600/are-homothetic-additively-separable-preferences-always-equivalent-to-ces/54601#54601 ).
The second to top "Related" question (https://economics.stackexchange.com/questions/6908/what-utility-functions-are-equivalent-to-additive-functions?rq=1) there contained a link to Ted Bergstrom's Lecture Notes on Separable Preferences, which answered this question in the affirmative.
The full proof is given in "Donald W. Katzner. Static Demand Theory. Macmillan, New York, 1970".