What is bigger: $\sqrt2^{\sqrt3^\sqrt3}$ or $\sqrt3^{\sqrt2^\sqrt2}$?

As the title implies: what is bigger $\sqrt2^{\sqrt3^\sqrt3}$ or $\sqrt3^{\sqrt2^\sqrt2}$. Specifically I am interested in working this out without actually calculating the values. So far I have tried applying order preserving operations on both and seeing if the comparison will become clearer but this has so far been unyieldy because I am stuck at the following point:

$\sqrt2^{\sqrt3^\sqrt3}$ or $\sqrt3^{\sqrt2^\sqrt2}$

$e^{\sqrt3^\sqrt3\ln\sqrt2}$ or $e^{\sqrt2^\sqrt2\ln\sqrt3}$

${\sqrt3^\sqrt3\ln\sqrt2}$ or ${\sqrt2^\sqrt2\ln\sqrt3}$

${\sqrt3^\sqrt3\ln2}$ or ${\sqrt2^\sqrt2\ln3}$

And at this point I have explored a few options but nothing has made it clear. Have I been pursuing the correct root (if you pardon the pun) and how should I proceed.

Update:

$\ln({\sqrt3^\sqrt3\ln2})$ or $\ln({\sqrt2^\sqrt2\ln3})$

$\frac{\sqrt3}{2}\ln3 +\ln({\ln2})$ or $\frac{\sqrt2}{2}\ln2 +\ln({\ln3})$

from Hot Weekly Questions - Mathematics Stack Exchange