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That is true. However, most of the worst-case examples I know are not uniformly positive. Nevertheless, I would be also happy with a counterexample for the estimate.
I should have written $C\lesssim 1$, I just wanted to say that the constant is small. However, I don't really care about the size of the constant. In my computations $L$ is normalized to have one's at the diagonal, but your normalization would also be fine.
