A search for pairs of boosted light Higgs bosons ($\mathrm{H}_1$) produced in supersymmetric cascade decays is performed in final states with small missing transverse momentum. The complete LHC Run II proton-proton collision data set is used, recorded with the CMS detector at a centre-of-mass energy of 13~\mathrm{TeV} and corresponding to an integrated luminosity of 137~\mathrm{fb}^{-1}. The search targets events where both $\mathrm{H}_1$ bosons decay into $\mathrm{b}\bar{\mathrm{b}}$ pairs that are reconstructed as large-radius jets using substructure techniques. No evidence is found for any excess of events beyond the background expectations of the standard model (SM). Results from the search are interpreted in the next-to-minimal supersymmetric extension of the SM, where a low-mass singlino leads to multi-step squark and gluino decays that can predominantly end with a boosted singlet-like $\mathrm{H}_1$ boson and a low-momentum singlino-like neutralino. Upper limits are set on the product of the squark or gluino pair-production cross section and the $\mathrm{b}\bar{\mathrm{b}}$ branching ratio of the $\mathrm{H}_1$ for a benchmark model with almost mass-degenerate light flavour squarks and gluinos. Under the assumption of an SM-like $\mathrm{H}_1\to\mathrm{b}\bar{\mathrm{b}}$ branching ratio, $\mathrm{H}_1$ bosons with masses in the range 40--120 GeV, arising from the decays of squarks or gluinos with a mass from 1200--2500 GeV, are excluded at the $95\%$ confidence level.

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