Does this sum $(1-\frac{1}{2^2})^{(1-\frac{1}{3^2})^{...^{(1-\frac{1}{p^2})}}}$ also related to Riemann zeta function?

Question

I'm interesting for iterated exponention sum of the form $(z_1)^{z_2)^{...^{z_k}}}$ such that $z_1$ and $z_2$, $z_k$ are differents real exponents , This kind of sum was studied by many Authors such as : Barraw (infinit exponential , Monthly 28 (1921) pp 141-143), For Euler product which it is related to Riemann zeta function we have this well known identity :$ \prod_{p\in \mathbb{P}} (1-p^{-s})=\frac{1}{\zeta(s)}$ , for $s=2$ This product gives $\frac{6}{\Pi^2}$ , Now it comes to my mind to do that product as exposnention sum as follow :$$(1-\frac{1}{2^2})^{(1-\frac{1}{3^2})^{...^{(1-\frac{1}{p^2})}}}$$ , Now I want to know what's equal this sum because obviously it is convergent since each exponent lie between $(0,1)$ and in the same time does this iterated exponention sum related to Riemann zeta function as Euler product for large enough prime ?