Close to the continuum limit, lattice QCD with mass-degenerate Wilson quarks can be described by Symanzik's effective continuum action,
which contains the dimension 5 operator, $m {\rm tr}(F_{\mu\nu}F_{\mu\nu})$. Its effect can be eliminated by an O($a m_{\rm q}$) rescaling of the bare lattice coupling constant.
Until recently, the corresponding improvement coefficient, $b_{\rm g}$, was only known perturbatively to 1-loop order and
an estimate of the remaining uncertainty is the dominant systematic error in the ALPHA collaboration's recent determination of $\alpha_s(m_Z)$ with the decoupling method~\cite{DallaBrida:2022eua}.
To remove this error we have determined$ b_{\rm g}$ non-perturbatively for the corresponding parameter range. We here briefly review improvement conditions for $b_{\rm g}$, perform a perturbative test and report on our non-perturbative results for $b_{\rm g}$.