On a characterization of probability distribution based on maxima of independent or max-independent random variables

Kotlarski (1978) proved a result on identification of the distributions of independent random variables $X,Y$ and $Z$ from the joint distribution of the bivariate random vector $(U,V)$ where $(U,V)=(\max (X,Z),\max (Y,Z))\mathrm{.}$ We extend this result to the case $(U,V)=(\max (X,a{Z}_1,b{Z}_2),\max (Y,c{Z}_1,d{Z}_2))$ where $X,Y,Z_1,Z_2$ are independent or max-independent random variables, $Z_1$ and $Z_2$ are identically distributed and $a,b,c,d$ are known positive constants.