Closed Grothendieck Construction

CAUTION: The Sender of this email is not from within Dalhousie. Hello, The equivalence of monoidal indexed categories and monoidal fibrations has been proved in "Monoidal Grothendieck Construction" by Joe Moeller and Christina Vasilakopoulou. (https://arxiv.org/abs/1809.00727) Has the analogous statement for closed structure been proved? It appears to be folklore, often assumed in categorical logic and elsewhere. It is likely that I simply haven't found the right reference. Any help is appreciated. Thank you, Christian Williams [For admin and other information see: http://www.mta.ca/~cat-dist/ ]