Convergence of Variable Base Power Tower $\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\ldots}}}$

Question

I’m curious if there is a heuristic method that can be used to solve what appears to be an elementary power tower where every increasing power is decreasing by a half integer. Numerical methods suggest it very slowly oscillates toward $0.67$ between $0.65$ and $0.69$. The power tower begins at $\frac{1}{2}$ and is raised to the $\frac{1}{n}$-th power after each iteration: $$\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\frac{1}{5}^\ldots}}}$$ I suspect that there is some way to transform this into some kind of infinite product, or another method that would resolve this, or refining the equation $\frac{W(\ln2)}{\ln2}$. Where $W(x)$ is the Lambert-W function. The given equation would compute the infinite power tower of $\frac{1}{2}$, so one would just find the difference of each successive power and weight it against the initial value of the sum, $\frac{W(\ln2)}{\ln2}$, such that it converges to the desired value by summation. Is there heuristic way to approach this problem?