Conjecture: Tree Evaluation is not in NL

Prove that the Tree Evaluation problem—given a complete d-ary tree of height h with b-bit leaf values and each internal node labeled by a function mapping d·b input bits to b output bits, decide the first bit of the root’s value—is not in NL (nondeterministic logspace).

Background

Tree Evaluation was introduced by Cook, McKenzie, Wehr, Braverman, and Santhanam and has been a focal problem in space-bounded complexity. A negative membership result (TE ∉ NL) would separate NL from P and yield stronger separations.

Despite significant algorithmic progress culminating in a space-efficient algorithm by Cook and Mertz, the original conjecture remains an explicit target, though recent work raises skepticism about its truth.