Call a poset locally countable if the set of predecessors of every member of the poset is countable. Is the following consistent?

There is no locally countable poset $P$ of size continuum such that every locally countable poset of size continuum embeds into $P$?

Remark: under ZFC+CH, there is such a universal locally countable poset: the Turing degrees. This is due to Sacks.