Asymptotic bounds of product of $\log(i)$

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$$\prod _{k=2}^n\left(\log_2k\right)$$

Can somebody help me with bounds of this expressions. I see only the rude measure:

$$\log_2n\le \prod _{k=2}^n\left(\log_2k\right)\le \left(\log_2n\right)^n$$