  Let $\Phi$ be a $C^*$-subalgebra of $L^\infty(\mathbb{R})$ and
$SO_{X(\mathbb{R})}^\diamond$ be the Banach algebra of slowly oscillating
Fourier multipliers on a Banach function space $X(\mathbb{R})$. We show that
the intersection of the Calkin image of the algebra generated by the operators
of multiplication $aI$ by functions $a\in\Phi$ and the Calkin image of the
algebra generated by the Fourier convolution operators $W^0(b)$ with symbols in
$SO_{X(\mathbb{R})}^\diamond$ coincides with the Calkin image of the algebra
generated by the operators of multiplication by constants.
