Let $Y$ be a contractible finite simplicial 2-complex.  

Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H\_1(X)=0$, $H\_2(X)=0$).  

Is $X$ contractible? (Equivalently, is $\pi\_1(X)$ trivial?)


I asked this question among others [here before](https://mathoverflow.net/questions/87045/is-the-euler-characteristic-of-aspherical-connected-2-complexes-at-most-1-no), but it remained unanswered.


Apparently this question is equivalent to the following one.


If $m < n$, $G = \langle a\_1,\dotsc,a\_m \mid r\_1 = \dotsb = r\_m =1 \rangle$ and $J=\langle a\_1,\dotsc,a\_n \mid r\_1 = \dotsb = r\_n =1 \rangle$ are two groups given by (balanced) presentations, the presentation of $G$ is a sub-presentation of the presentation of $J$, and $J$ is trivial, is $G$ trivial? (It can be shown that the abelianization of $G$ is trivial.)



Are acyclic subcomplexes of finite contractible 2-complexes contractible?
According to Sergei Ivanov, *On balanced presentations of the trivial group*, Invent. Math. **165**, 525--549 (2006), for $n=m+1$, this is a particular case of Kervaire–Laudenbach conjecture which is also open, and according to one of Klyachko's results, the negative answer in this special case would imply the negative answer to Whitehead's asphericity conjecture.


It seems however that nothing is known in general, not even equivalence with the Kervaire–Laudenbach conjecture. 


