Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the identity by a single off-diagonal element. The question whether $SL_2(R) = E_2(R)$ is a problem of S. Bachmuth and H. Y. Mochizuki [2]. It is also an open problem in [3, MA1] and a conjecture in [4].

I am very curious about the status of this problem/conjecture or any progress on it.

**A complement prompted by the first answer:** P. M. Cohn proved that $SL_2(R) \neq E_2(R)$ for $R = \mathbb{Z}[X]$ by considering the matrix $\begin{pmatrix} 1 + 2X & 4 \\ -X^2 & 1 - 2X \end{pmatrix}$ after his [Lemma 8.4, 1]. This matrix is easily seen to sit in $E_2(\mathbb{Z}[X^{\pm 1}])$ by cancelling one coefficient and then using Whitehead's lemma. P. M. Cohn also proved using [Proposition 7.2, 1] that the matrix $\begin{pmatrix} 1 + XY & X^2 \\ -Y^2 & 1 - XY \end{pmatrix}$ doesn't lie into $E_2(k[X, Y])$ for any field $k$. This matrix is usually referred to as *Cohn's matrix* and admits the following generalization discussed by T. Y. Lam and T. Dorsey [Example VI.3.5, 5]:
$C_{m, n}(X, Y) = \begin{pmatrix} 1 + XY & X^n Y^m \\ (-1)^r X^m Y^n & 1 - XY + X^2 Y^2 - \cdots + (-1)^r X^r Y^r \end{pmatrix}$ where $m, n \ge 0$ and $r = m + n - 1 \ge 0$. The first answer's author legitimately asks whether $C_{0, 3}(X - 1, p)$ lies into $E_2(\mathbb{Z}[X^{\pm 1}])$ for $p$ a prime number (it is known that $C_{m, n}(x, y) \in E_3(R)$ for any elements $x, y$ in a commutative ring $R$).

[1] "On the structure of the $GL_2$ of a ring", P. M. Cohn, 1966.

[2] "$E_2 \neq SL_2$ for most Laurent polynomial rings", S. Bachmuth, H. Y. Mochizuki, 1982.

[3] "Open problems in combinatorial group theory", G. Baumslag et al., 2000.

[4] "On finite and elementary generation of $SL_2(R)$", P. Abramenko, 2007, http://arxiv.org/abs/0808.1095.

[5] "Serre's problem on projective modules", T. Y Lam, 2006.