Requirement, No, only uplinks are REQUIRED to register.
The big discussion here lately has been about registering downlinks since the FCC wants to auction off (some of) the C-band to Terrestrial wireless providers. It is optional, but recommended, at a cost of 435-1600ish depending on wether you are comfortable filing the paperwork yourself or not and wether you do frequency co-ordination along with it. Frequency co-ordination is optional, but likely the only way to ensure any protection from future interference.
From: Joel Force [mailto:firstname.lastname@example.org]
Sent: Friday, June 22, 2018 7:08 AM
To: CRTech <email@example.com>
Subject: [CRTech] FCC Dish Registration
Does anyone know if a 3.1 meter satellite dish, using Unity 4000, Moody Radio and Amb-OS satellite receivers, falls under the C-band registration requirement by the FCC?
I noticed in the Unity 4000 manual that it is considered an L-Band receiver.