Search for Quark Deconfinement: Strangeness Production in $\mathrm{pp}$, $\mathrm{dd}$, $p\alpha$, and $\alpha \alpha$ Collisions at $\sqrt{s_{\mathrm{NN}}}=31.5\mathrm{} \mathrm{and} 44$ GeV

Relative yields of $\phi$ and charged $K$ mesons are measured as functions of charged multiplicity in the central rapidity region for $\mathrm{pp}$, $\mathrm{dd}$, $p\alpha$, and $\alpha \alpha$ collisions at $\sqrt{s_{\mathrm{NN}}}=3.15\mathrm{} \mathrm{and} 44$ GeV. No anomalous strange-particle production is observed in $\alpha \alpha$ reactions relative to $\mathrm{pp}$ even at large multiplicities corresponding to events comprising less than 3.6×${10}^{-4\mathrm{}}$ of the inclusive cross section.