We present progress towards first continuum estimates of second-order conserved-charge cumulants in $(2+1)$-flavor QCD with M\"obius domain-wall fermions at the physical quark-mass point. The analysis is based on ensembles with temporal extents $N_\tau=12$ and $16$ and aspect ratio $N_\sigma/N_\tau=3$, which allow us to investigate lattice cutoff effects in the temperature range $135$--$205~\mathrm{MeV}$. We study the temperature dependence of $\chi_2^B$, $\chi_2^Q$, $\chi_2^S$, $\chi_{11}^{BQ}$, $\chi_{11}^{BS}$, and $\chi_{11}^{QS}$, and compare the results with HRG model calculations and with results obtained using staggered fermion discretizations. We also present first continuum estimates for selected second-order cumulants. At the present level of statistics, observables that do not involve derivatives with respect to the baryon chemical potential provide the cleanest basis for an initial continuum-estimate analysis, while the remaining second-order cumulants are significantly more challenging and are left for future work.