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Let $I$ be a homogeneous ideal in the polynomial ring $K[x_1,\cdots, x_n]$. The asymptotic generator degree of $I$ is defined to be the minimal number $d$ such that $I$ is integral over $I_{\leq d}$. In the case that $I$ is generated in a single degree, the asymptotic generator degree and the common degree of generators of $I$ coincide. My question is: In the case that $I$ is a monomial ideal, is it true that the asymptotic generator degree is equal to the maximum degree of a minimal generator of $I$? Any reference or hint to the answer is appreciated!

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