Non-perturbative determination of improvement $b$-coefficients in $N_f=3$

We present our preliminary results of the non-perturbative determination of the valence mass dependent coefficients $b_{\mathrm{A}}-b_{\mathrm{P}}$ and $b_{\mathrm{m}}$ as well as the ratio $Z_{\mathrm{P}}{Z}_{\mathrm{m}}\mathrm{/}{Z}_{\mathrm{A}}$ entering the flavour non-singlet PCAC relation in lattice QCD with $N_f=3$ dynamical flavours. We apply the method proposed in the past for quenched approximation and $N_f=2$ cases, employing a set of finite-volume ALPHA configurations with Schr"odinger functional boundary conditions, generated with $O(a)$ improved Wilson fermions and the tree-level Symanzik-improved gauge action for a range of couplings relevant for simulations at lattice spacings of about $0.09$fm and below.