There have been multiple conjectures of this type - seemingly motivated, commonly believed, yet false - about the structure of the partial order of Turing degrees of c.e. sets. Two in particular were due to Shoenfield:

• In 1963, he conjectured that, given any finite poset $P$ which embeds (via $f$) into the c.e. degrees preserving $0, 1$, and $\vee$, and $P'\supseteq P$ has the same maximal and minimal elements, and l.u.b.s, as $P$, then $P'$ embeds into the c.e. degrees via an embedding extending $f$.

This was refuted by the construction of a minimal pair of c.e. degrees, that is, a pair of noncomputable c.e. degrees $\underline{a}$, $\underline{b}$ such that no noncomputable set is computable in both $\underline{a}$ and $\underline{b}$.

• Eleven years later, Shoenfield conjectured that all finite lattices were embeddable into the c.e. degrees in a way that preserved 0.

Manuel Lerman counter-conjectured that the lattice $S_8$ was not so embeddable; this was proved by Lachlan and Soare six years later.

The motivation behind both conjectures was the intuition that the c.e. degrees were a nicely behaved structure; in particular, I think it was believed that the poset of degrees c.e. in and above a given $\underline{d}$ should be isomorphic to the poset of c.e. degrees, that the theory of the c.e. degrees is decidable, that the poset is $\aleph_0$-categorical, etc., and all of these turned out to be false.