In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each exponent level but different.

I suspect this case is not possible, but haven't been able to formally rule it out. So to make it concrete, is it possible that two power towers $$ S = p^{q^{\Large r^{\Large s^t}}} \quad T = v^{w^{\Large x ^ {\Large y^z}}} $$ with $1<p,q,r,s,t,v,w,x,y,z<N=2^{10}$ satisfy $$ S<T<2S $$ ?

If not and ideally can we bound the distance in terms of $N$? As @fedja commented, without a constraint we can always let $M=q^{r^{s^t}}=w^{x^{y^z}}$ and find $(v/p)^M<2$ with $p$ large enough.