Dilaton Weyl multiplets for $N=3$ conformal supergravity in four dimensions

We construct a dilaton Weyl multiplet for $N=3$ conformal supergravity in four dimensions. We couple an on-shell vector multiplet to the standard Weyl multiplet and use the field equations of the vector multiplet to replace some of the components of the auxiliary fields of the standard Weyl multiplet with the fields of the vector multiplet and some dual gauge fields. The R-symmetry of the multiplet is $SU(2)\times U(1)\times U(1)$. Furthermore, we gauge fix one of the two $U(1)$ symmetries and rewrite the result for the dilaton Weyl multiplet with $SU(2)\times U(1)$ R-symmetry.