The rational-generating-function conjecture for motivic partition errors

For nonnegative integers $k$ and $d$, let $e_{k,d}$ denote the numerical discrepancy associated with the partition/Hilbert-scheme comparison, and let $\mathsf M_k(t)$ be a polynomial if it exists. Rational-error conjecture. For every $k\geq 0$, there exists a polynomial $\mathsf M_k(t)\in{\mathbb Q}[t]$ such that

$$\sum_{i\geq 0}e_{6+k+i,4+i}t^i= \frac{\mathsf M_k(t)}{(1-t)^{2k+3}\prod_{i=0}^{k}(1-(2+i)t)^{k+1-i}}.$$

The conjecture proposes a uniform rational form for these discrepancy sequences, extending the limited computational knowledge of their structure.