Uniform-child toric ideals are quadratically generated

Let $T$ be a rooted tree and let $I_T$ be the toric ideal associated to the homogeneous binary Markov model. Suppose either that $T$ is binary, or more generally that every non-leaf node of $T$ has the same number $d$ of children, with $d\geq 2$. The toric ideal $I_T$ should be generated in degree $2$. This conjecture is based on Markov-basis computations for binary trees with at most 11 nodes and selected larger examples; it concerns generation by quadrics, not the existence of a quadratic Gröbner basis.