Let $R=k[x_1,\ldots,x_n]$ be a graded ring and $S,T,U$ be monomials ideals. $reg(S)=max\{j-i \backslash \beta_{i,j}(S) \neq 0\}$.

Assume $S+T=U$

**prove \disprove :** $reg(S+T^2) \leq reg(U^2)$.

We can see $reg(S+T^2) \leq reg(S)+reg(T^2)-1 < reg(S)+reg(T^2)$ by By Herzog result, see Corollary 3.2