Is it true $\sqrt{2^{3^{-5^{-7^{11^{13^{-17^{-19^{23^{29^{-31^{-37^{41^{\ldots}}}}}}}}}}}}}} =\sqrt{2}$?

Convergence of your tower to $2$ is equivalent to convergence of $3^{-5^{-7^{11^\ldots}}}$ to $1$, which is equivalent to convergence of $5^{-7^{11^{13^\ldots}}}$ to $0$, which is equivalent to convergence of $7^{11^{13^{-17^\ldots}}}$ to $\infty$, which is equivalent to convergence of $11^{13^{-17^{-19^\ldots}}}$ to $\infty$, which is equivalent to convergence of $13^{-17^{-19^{23^\ldots}}}$ to $\infty$, which is equivalent to convergence of $17^{-19^{23^{29^\ldots}}}$ to $-\infty$, which is absurd.