Let $Z$ be a simply connected, two dimensional simplicial complex.

Let $X\subset Z$ be a finite subcomplex with nontrivial $\pi_{1}$.

Must there exist a finite, simply connected subcomplex $Y\subset Z$ such that $Y\supset X$?

(Motivation: the fact that Whitehead conjecture remains unproven indicates that there are probably some very weird things that can happen in two dimensional complexes. This question attempts to locate some of these pathologies.)