Characterization of pairs invariant in both types

Let

$$m(a,b)=\dfrac{a+b}{2},\qquad M(a,b,c)=\dfrac{a+b+c}{3}.$$

A pair of means $(\overline{m},\overline{M})$ is said to satisfy both $(\overline{M},\overline{m})_{1}$ and $(\overline{M},\overline{m})_{2}$ when it is invariant in both types defined in the source.

Both-types invariance conjecture. The only such pairs are the quasi-arithmetic transforms

$$\overline{m}(a,b)=h^{-1}(m(h(a),h(b))),$$ $$\overline{M}(a,b,c)=h^{-1}(M(h(a),h(b),h(c))),$$

where $h(u)$ is monotonic on $(0,\infty)$.

The conjecture is motivated by the calculation showing that the invariant logarithmic mean is type 1 invariant but not type 2 invariant with respect to the logarithmic mean. The source offers this as a belief and does not provide a proof or resolution.