A PC’s familiar would vanish as it attempted to enter an anti-magic field, thereby blocking any communication between Ichabod and the Imp.
Antimagic field; This area is divorced from the magical energy that suffuses the
multiverse. Within the Sphere, Spells can't be cast, summoned
creatures disappear, and even Magic Items become mundane...
Spells and other magical Effects, except those created by an artifact or a deity, are suppressed in the Sphere and can't protrude into it. PHB. 210
Since the Imp is summoned by Ichabod via magic; (Find Familiar spell) the anti-magic field will cause the Imp to disappear.
There is an unusual case with the variant familiar rules intended for NPC’s.
These Imp familiars are acquired by contract and are not summoned via magic.
Familiar. The imp can enter into a contract to serve another creature as a familiar, forming a telepathic bond with its wi lling master.
While the two are bonded, the master can sense what the imp senses as
long as they are within l mile of each other. MM. 69
Although an Imp under contract will not disappear entering an anti-magic field, their telepathy may cease to function.
The monsters manual advises that telepathy is a magical ability, as such, telepathy should be suppressed by anti-magic.
Telepathy Telepathy is a magical ability that allows a monster to communicate
mentally with another creature within a specified range. MM. 10