Let $F$ be a free group and $k$ be a field. If $x$ is an element of the group algebra $k[F]$ that is not a unit (equivalently, that is not a nonzero scalar multiple of an element of $F$), must the 2-sided ideal $k[F]\,x\,k[F]$ generated by $x$ be proper? In other words, is it true that for all $y_1,\dots, y_n, z_1,\dots, z_n \in k[F]$, we have $$y_1 x z_1 + y_2 x z_2 +\dots + y_n x z_n \neq 1 ?$$

This question was asked by George Bergman. For the application he has in mind, it would actually be enough to show that if $a$ is one of the free generators of $F$, and $x$ is an element of $k[F]$ whose support consists of elements of $F$ having total degree $0$ in $a$, then $1\notin k[a]\,x\,k[F]$. (For example, $x$ might be $b + abaca^{-2}$, where $b$ and $c$ are two other members of a free generating set for $F$.) Equivalently, he would like to know that the right ideal of $k[F]$ generated by the conjugates of $x$ by the non-negative powers of $a$ is a proper ideal.