Determinantal inequality threshold conjecture for the broken k-diamond partition function

Let $\Delta_k(n)$ denote the broken $k$-diamond partition function. For a sequence, its order-$m$ determinantal inequality is the nonnegativity of the Toeplitz minor

$$\det\bigl(\Delta_k(n-i+j)\bigr)_{1\leq i,j\leq m}.$$

Determinantal threshold conjecture. For $k=1$ or $2$ and $1\leq m\leq14$, one has

$$\det\bigl(\Delta_k(n-i+j)\bigr)_{1\leq i,j\leq m}\geq0$$

for $n\geq\mathbf N_{\Delta_k}(m)$, with thresholds

$$\begin{array}{c|rrrrrrrrrrrrrr} m&1&2&3&4&5&6&7&8&9&10&11&12&13&14\\ \mathbf N_{\Delta_1}(m)&1&1&20&84&194&362&594&890&1258&1700&2218&2818&3498&4264\\ \mathbf N_{\Delta_2}(m)&1&1&18&72&168&308&506&762&1082&1464&1914&2436&3028&3696 \end{array}$$

The statement is posed among the paper's open problems as numerical evidence for eventual determinantal positivity.