In Nowak (1989), a modular $\rho$ on a vector lattice is defined by the following properties

(N1) $\rho(x)=0\implies x=0$;

(N2) $\lvert x\rvert \le \lvert y\rvert\implies \rho(x) \le \rho(y)$;

(N3) $\rho(x\vee y )\le \rho(x)+\rho(y)$ for all $x\ge 0, y\ge 0$;

(N4) $\rho(\lambda x)\to 0$ if $\lambda \to 0$

This supposedly implies $$ \rho(\alpha x+\beta y)\le \rho( x)+\rho(y) \text{ for all } \alpha,\beta\ge0,\alpha+\beta=1,$$ which features in the definition of modular in Musielak and Orlicz (1959).

**I don't see how this follows. NB: convexity is not assumed.**