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Given a topological dynamical system $(X,T)$ and an arithmetic function$\boldsymbol{u}\colon\mathbb{N}\to\mathbb{C}$, we study the strong MOMOproperty (relatively to $\boldsymbol{u}$) which is a strong version of$\boldsymbol{u}$-disjointness with all observable sequences in $(X,T)$. It isproved that, given an ergodic measure-preserving system$(Z,\mathcal{D},\kappa,R)$, the strong MOMO property (relatively to$\boldsymbol{u}$) of a uniquely ergodic model $(X,T)$ of $R$ yields all otheruniquely ergodic models of $R$ to be $\boldsymbol{u}$-disjoint. It follows thatall uniquely ergodic models of: ergodic unipotent diffeomorphisms onnilmanifolds, discrete spectrum automorphisms, systems given by somesubstitutions of constant length (including the classical Thue-Morse andRudin-Shapiro substitutions), systems determined by Kakutani sequences areM\"obius (and Liouville) disjoint. The validity of Sarnak's conjecture impliesthe strong MOMO property relatively to $\boldsymbol{\mu}$ in all zero entropysystems, in particular, it makes $\boldsymbol{\mu}$-disjointness uniform. Theabsence of strong MOMO property in positive entropy systems is discussed and,it is proved that, under the Chowla conjecture, a topological system has thestrong MOMO property relatively to the Liouville function if and only if itstopological entropy is zero.

M\"obius disjointness for models of an ergodic system and beyond