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

Let $\Delta_k(n)$ denote the broken $k$-diamond partition function, and let $L_m(a_n)$ denote the discrete Laguerre expression of order $m$:

$$L_m(a_n)=\frac{1}{2}\sum_{r=0}^{2m}(-1)^{r+m}{2m\choose r}a_{n+r}a_{n+2m-r}.$$

Laguerre threshold conjecture. For $k=1$ or $2$ and $1\leq m\leq14$, the sequence $\Delta_k(n)$ satisfies the Laguerre inequality of order $m$, namely $L_m(\Delta_k(n))\geq0$, for $n\geq N_{\Delta_k}(m)$, where

$$\begin{array}{c|rrrrrrrrrrrrrr} m&1&2&3&4&5&6&7&8&9&10&11&12&13&14\\ N_{\Delta_1}(m)&1&12&53&132&251&420&639&912&1245&1636&2091&2612&3201&3858\\ N_{\Delta_2}(m)&1&10&45&106&211&354&539&774&1059&1398&1781&2240&2749&3318 \end{array}$$

These are computationally proposed thresholds for extending the proved low-order Laguerre inequalities.