Unimodality conjecture for Garvan k-rank functions

Let $N_k(m,n)$ denote the number of partitions of $n$ with Garvan $k$-rank $m$. Let ${\bf 1}_{E}$ be the indicator of a condition $E$, and define

$$n_u(k)=(k+1)+36\,{\bf 1}_{k=2}+6\,{\bf 1}_{k=3}.$$

Garvan k-rank unimodality conjecture. For each integer $k\ge2$ and integer $n\ge n_u(k)$, the sequence

$$\left(N_k(m,n)\right)_{|m|\le n-k}$$

is unimodal. The source reports computational verification for $k\in\{2,3,4,5,6,7,8,9,10\}$ and $n\le1000$, while the preceding theorem only establishes unimodality for sufficiently large $n$.