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The use of Heavy Quark Effective Theory (HQET) on the lattice as an approachto B-physics phenomenology is based on a non-perturbative matching of HQET toQCD in finite volume. As a first step to apply the underlying strategy in thethree-flavor ($N_f = 2+1$) theory, we determine the renormalization constantand improvement coefficients relating the renormalized current and subtractedquark mass of (quenched) valence quarks in $\mathcal{O}(a)$ improved $N_f=3$lattice QCD. We present our strategy and first results for the relevantparameter region towards weak couplings along a line of constant physics, whichcorresponds to lattice resolutions $a\leq 0.02\,$fm and fixes the physicalextent of the matching volume to $L\approx 0.5\,$fm.

$\mathcal{O}(a)$ improved quark mass renormalization for a non-perturbative matching of HQET to three-flavor QCD