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Yes, and moreover, for arbitrary convex function $f$ defined on the interval between $a_1+\dots+z_1$ and $a_n+\dots+z_n$ we have $$f(a_1 + b_{\sigma(1)}+\cdots +z_{\tau(1)}) + \cdots + f(a_n + b_{\sigma(n)}+ \cdots + z_{\tau(n)}) \leqslant f(a_1 + b_1 + \cdots + z_1) + \cdots + f(a_n + b_n + \cdots + z_n).$$ This follows immediately from Karamata's inequality: for every $k=1,\dots,n$ the sum of any $k$ arguments of $f$ in LHS does not exceed the sum of the last $k$ arguments of $f$ in RHS, and for $k=n$ these sums are equal.