Broadhurst–Kreimer conjecture on indecomposable multiple zeta values

Let $D_{n,k}$ denote the number of multiple zeta values of weight $n$ and depth $k$ that are not reducible to multiple zeta values of lesser depth. Broadhurst–Kreimer conjecture. The numbers $D_{n,k}$ are determined by

$$1-\frac{x^3y}{1-x^2}+\frac{x^{12}y^2(1-y^2)}{(1-x^4)(1-x^6)}=\prod_{n\ge3}\prod_{k\ge1}(1-x^ny^k)^{D_{n,k}}.$$

This conjecture predicts the generating series for the depth-graded indecomposable part of the algebra of multiple zeta values, thereby describing which values cannot be reduced to lower-depth ones. The supplied source gives no evidence of a resolution, so its status is left open.