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We update the full set of $\bar{B}_s\rightarrow K$ form factors usinglight-cone sum rules with an on-shell kaon. Our approach determines therelevant sum rule parameters -- the duality thresholds -- from a Bayesian fitfor the first time. Using a modified version of the Boyd-Grinstein-Lebedparametrisation, we combine our sum rule results at low momentum transfer $q^2$with more precise lattice QCD results at large $q^2$. We obtain a consistentdescription of the form factors in the full $q^2$ range. Applying these resultsto a recent LHCb measurement of branching ratios for the decays $B_s^0 \to\lbrace K^-, D_s^-\rbrace \mu^+\nu_\mu$, we determine the ratio ofCabibbo-Kobayashi-Maskawa elements $$ \notag\left|\frac{V_{ub}}{V_{cb}}\right|_{q^2<7\; \textrm{GeV}^2} = 0.0681\pm 0.0040\quad \text{and} \quad \left|\frac{V_{ub}}{V_{cb}}\right|_{q^2>7\;\textrm{GeV}^2} = 0.0801\pm 0.0047 \ , $$ which are mutually compatible atthe $1.9\sigma$ level. We further comment on the sensitivity to Beyond theStandard Model effects through measurements of the shape of $B_s^0 \to K^-\mu^+\nu_\mu$ decays, in light of recent limits on such effects from otherexclusive $b\to u\ell\nu$ processes.

New determination of $|V_{ub}/V_{cb}|$ from $B_s^0\to \lbrace K^-, D_s^- \rbrace \mu^+\nu$