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We initiate the analysis of the Kaluza--Klein mass spectrum of massive IIAsupergravity on the warped AdS$_6 \times_w S^4$ background, by deriving thelinearised equations of motion of bosonic and fermionic fluctuations, anddetermining the mass spectrum of those of spin-2. The spin-2 modes are given interms of hypergeometric functions and a careful analysis of their boundaryconditions uncovers the existence of two branches of mass spectra, bounded frombelow. The modes that saturate the bounds belong to short multiplets which weidentify in the representation theory of the $\mathfrak{f}(4)$ symmetrysuperalgebra of the AdS$_6 \times_w S^4$ solution.

Perturbing AdS$_6 \times_w S^4$: linearised equations and spin-2 spectrum