Eigenvariety for partially classical Hilbert modular forms

For each subset of primes in a totally real field above a rational prime $p$, there is the notion of partially classical Hilbert modular forms, where the empty set recovers the overconvergent forms and the full set of primes above $p$ yields classical forms. Given such a set, we $p$-adically interpolate the classical modular sheaves to construct families of partially classical Hilbert modular forms with weights varying in appropriate weight spaces and construct the corresponding eigenvariety, generalizing the construction of Andreatta, Iovita, Pilloni, and Stevens.