Non-universality conjecture for height-one locally countable Borel quasi-orders

A quasi-order $(P,\leq_P)$ is a reflexive and transitive relation, and its height is the length of the longest strictly decreasing chain, equivalently the height of the partial order obtained by quotienting by $x\sim_P y$ when $x\leq_P y$ and $y\leq_P x$. A quasi-order on a standard Borel space is locally countable when every element has at most countably many strict predecessors, and it is Borel when its underlying relation is Borel. Non-universality conjecture. There is a locally countable Borel quasi-order of height one which is not Borel reducible to Turing reducibility. This would show that the height-one quasi-order analogue of Kechris's universality conjecture fails; the paper contrasts it with its positive result for height-two locally countable Borel partial orders.