The integer-scaling criterion for C-realizability

Let $\{\frac{\lambda_1}{\alpha_1},\ldots,\frac{\lambda_n}{\alpha_n}\}$ be a multiset of rational numbers, and let $p=\operatorname{lcm}(\alpha_1,\ldots,\alpha_n)$. Here $C_{\mathbb{Q},3}$-realizability and $C_{\mathbb{Z},3}$-realizability refer to the corresponding realizability criteria for the Nonnegative Inverse Eigenvalue Problem.

Integer-scaling criterion. If $\{\frac{\lambda_1}{\alpha_1},\ldots,\frac{\lambda_n}{\alpha_n}\}$ is $C_{\mathbb{Q},3}$-realizable, then it is $C_{\mathbb{Q},3}$-realizable if and only if

$$\left\{p\cdot\frac{\lambda_1}{\alpha_1},\ldots,p\cdot\frac{\lambda_n}{\alpha_n}\right\}$$

is $C_{\mathbb{Z},3}$-realizable.

This criterion would remove the dependence on the particular sequence of realizability rules used to establish $C_{\mathbb{Q},3}$-realizability, reducing the question to an integer-scaled multiset whose scaling factor depends only on the denominators of the entries. The supplied text does not establish the equivalence, so its resolution is unclear.