Center-generation conjecture for the universal Drinfeld–Yetter algebra

Let $\mathfrak{U}_{\mathsf{DY}}^1$ be the universal Drinfeld–Yetter algebra, with unit-labelled element $r_1^{\mathsf{id}}$. Center-generation conjecture. The center of $\mathfrak{U}_{\mathsf{DY}}^1$ is generated by $r_1^{\mathsf{id}}$. This predicts a complete description of the central subalgebra of the universal Drinfeld–Yetter algebra; the supplied text gives no evidence of a proof or disproof.