Stabilization conjecture for near-perfect powers of colored partition numbers

Let $p_r(n)$ denote the $r$-colored partition function, and define $L_{p_r}(d)$ and $M_{p_r,k}(d)$ analogously to the corresponding quantities for the partition and overpartition functions. The colored-partition stabilization conjecture. For each non-negative integer $d$, there is a positive integer $N_d$ such that for all $k\geq N_d$,

$$M_{p_r,k}(d)=L_{p_r}(d).$$

The source proposes this as an analogue of the Merca–Ono–Tsai stabilization conjecture and gives no resolution.