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., ${\gamma }_S\le 1$. Exact baryon number conservation is found to be even more important than that of strangeness in the canonical suppression picture at the CERN Large Hadron Collider, in contrast to intermediate and low collision energies. The model is applied to $p\text{−}p$, $p$-Pb, and Pb-Pb data of the ALICE Collaboration. A chemical equilibrium CSM with a fixed $T_{\mathrm{ch}}=155$ MeV describes the trends seen in most yield ratios. However, this “vanilla” version of CSM predicts an enhancement of the $\varphi /\pi$ 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 ${\gamma }_S\le 1$ is accepted. Both the canonical suppression and the strangeness undersaturation effects are small at $d{N}_{\mathrm{ch}}/d\eta \gtrsim 100$, but they do improve substantially the description of hadron yields in $p\text{−}p$ collisions, in particular the $\Omega$ yields. A possibility to constrain the rapidity correlation volume using net-proton fluctuation measurements is pointed out.