The generating-function conjecture for multiple Clausen value irreducibles

For weight $n$ and depth $k$, let $P(n,k)$ denote the number of irreducible multiple Clausen values such that a rational basis at that weight and depth is formed from a minimum number of irreducibles, augmented by products of irreducibles of lesser weight and depth. Generating-function conjecture. This weight-and-depth filtration is generated by

$$\prod_{n>1,\,k>0}(1-x^ny^k)^{P(n,k)}=1-\frac{x^2y}{1-x}.$$

This conjecture refines the preceding dimension-counting proposal by tracking both weight and depth. The paper gives it as a pattern suggested by computational experiments; no proof or resolution is supplied in the source.