Castelnuovo- Mumford regularity properties

Question

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

Answer 1

This example is due to Aldo Conca:

$R = k[x_1,x_2,x_3], S = (x_1^3x_2,x_1x_2^3,x_2^4,x_1^2x_2^2x_3^5), T = (x_1^4), U = S+T$. Then $\text{reg} (S+T^2) = 9$, while $\text{reg} U^2 = 8$.