Let $p$ be a prime, and $F$ a $p$-adic Hida family of ordinary modular forms (of some tame level $N \ge 1$). I'd like to know whether, for $q$ a prime factor of $N$, the actions of the Atkin--Lehner involutions $W_q$ on the classical specializations of $F$ interpolate $p$-adically. (Equivalently, I'd like to know if there's an operator $W_q$ on the space of ordinary $\Lambda$-adic cusp forms of tame level $N$, which corresponds with its classical namesake under the specialization map at each classical weight.)

I've convinced myself that this is true -- I can bash out a conditional proof modulo some plausible-looking integrality statements -- but I'd be happier if I had a complete proof. Does anyone know a reference for this? (I've consulted various papers of Hida without finding anything.)

I'd also be interested in the corresponding question for Coleman families of non-zero slope.