On the equality problem of two-variable Bajraktarevi\'c means under first-order differentiability assumptions

The equality problem of the two-variable Bajraktarevi\'c means can be expressed as the functional equation $$\left(\frac{f}{g}\right)^{-1}\bigg(\frac{f(x)+f(y)}{g(x)+g(y)}\bigg) =\left(\frac{h}{k}\right)^{-1}\bigg(\frac{h(x)+h(y)}{k(x)+k(y)}\bigg)\qquad(x,y\in I),$$ where $I$ is a nonempty open real interval, $f,g,h,k:I\to{\mathbb R}$ are continuous functions, $g$, $k$ are positive and $f/g$, $h/k$ are strictly monotone. This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. Additional and new characterizations of this equality problem have been found recently by Losonczi, P\'ales and Zakaria under the same regularity assumptions in 2021. In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability.