nt.number theory - S-unit equation and small sets of places

Let $K$ be a number field, and let $S_x$ denote the set of primes of norm at most $x$. Is it possible to find a smaller set of places $T_xsubset S_x$ so that a lot of the solutions of the $S_x$-unit equation $a+b=1$ for $a,bin S_x$ are solutions of the $T_x$-unit equation?

Here's a possible precise statement (although I'd be interested in other formulations as well): Does there exist a constant $0<c<1$, depending on $K$ (but not $x$), so that for each $x$, there is a $T_xsubset S_x$ with $|T_x|lesqrt{|S_x|}$ so that the number of solutions to the $T_x$-unit equation is at least $c$ times the number of solutions of the $S_x$-unit equation?

I'm interested in this mostly by analogy: Bellabas and Gangl have a bound for the set of places of a number field one must check in order to compute $K_2$ of the ring of integers. It would be interesting to know if one could at least get a pretty good approximation for $K_2$ by looking at a much smaller set of places.