The equality problem in the class of conjugate means

Let ${Isubsetmathbb{R}}$ be a nonempty open interval and let ${L:I^2to I}$ be a fixed strict mean. A function ${M:I^2to I}$ is said to be an L-conjugate mean on I if there exist ${p,qin{]}0,1]}$ and a strictly monotone and continuous function φ such that $$M(x,y):=varphi^{-1}(pvarphi(x)+qvarphi(y)+(1-p-q)varphi(L(x,y)))=:L_varphi^{(p,q)}(x,y),$$ for all ${x,yin I}$ . Here L(x, y) is a fixed quasi-arithmetic mean. We will solve the equality problem in this class of means.