Characterization of reciprocal functions by independence of interpolation orders

Let $M_{f,m_{1},m_{2}}$ be the means defined in the paper, with $m_{1}$ and $m_{2}$ denoting the interpolation orders, and let $f$ be a function on the relevant domain. Reciprocal-function characterization problem. Show that the only function for which $M_{f,m_{1},m_{2}}$ is independent of $m_{1}$ and $m_{2}$ is

$$f(x)=\tfrac{C}{x}.$$

The paper proves that the reciprocal function produces the harmonic mean independently of the two interpolation orders. The displayed statement asks whether this is the only such function and is therefore an open problem rather than an asserted conjecture.