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We describe in detail the derivation of a superconformal off-shellformulation of the alternative $\mathcal{N}=2$, $d=5$ ungauged supergravity ofNishino and Rajpoot, coupled to $n$ Abelian vector multiplets, using a generaldilaton-Weyl multiplet. We generalize the vector multiplet coupling availablein the literature and show under which assumptions that the scalar manifoldreduces to the known case of SO(1,1) $\times$ SO(1,n)/SO(n). As an applicationof the formalism we propose generalized vector multiplet coupled highercurvature terms, whose construction we sketch briefly.

$\mathcal{N}=2$ dilaton-Weyl multiplets in 5D and Nishino-Rajpoot supergravity off-shell