The system-size dependence of hadrochemistry at vanishing baryon density is considered within the canonical statistical model (CSM) with local exact conservation of three conserved charges, allowing for a possibility of strangeness undersaturation, i.e. $\gamma_S \leq 1$. Exact baryon number conservation is found to be at least as important as strangeness conservation in the canonical suppression picture at the LHC. This is in contrast to intermediate and low collision energies, where the canonical effects on particle yields are dominated by strangeness conservation. The model is applied to p-p, p-Pb, and Pb-Pb data of the ALICE collaboration. A chemical equilibrium CSM describes the trends seen in most yield ratios. However, the CSM predicts an enhancement of the $\phi/\pi$ ratio at smaller multiplicities, in stark contrast to the suppression seen in the data. The data are described with a 15% relative accuracy level whence a multiplicity dependence of both the temperature and the strangeness saturation parameter $\gamma_S \leq 1$ is accepted. Both the canonical suppression and the strangeness undersaturation effects are small at $d Nch / d \eta \geq 100$, but they do improve substantially the description of hadron yields in p-p collisions, in particular the $\Omega$ yields.