Eventual three-term representations by generalized pentagonal numbers

Let $p_m(x)$ and $\bar p_m(x)$ denote the generalized polygonal-number functions used in the paper, and let $\mathbb N$ denote the nonnegative integers. Eventual representation conjecture. The following four inclusions hold:

$$\{p_5(x)+p_5(y)+p_5(z):x,y,z\in\mathbb N\}\supseteq\{n\in\mathbb N:n>33066\},$$ $$\{\bar p_5(x)+\bar p_5(y)+\bar p_5(z):x,y,z\in\mathbb N\}\supseteq\{n\in\mathbb N:n>24036\},$$ $$\{p_6(x)+p_6(y)+p_6(z):x,y,z\in\mathbb N\}\supseteq\{n\in\mathbb N:n>146858\},$$ $$\{\bar p_6(x)+\bar p_6(y)+\bar p_6(z):x,y,z\in\mathbb N\}\supseteq\{n\in\mathbb N:n>138158\}.$$

These conjectures assert explicit eventual universality for four ternary sums of generalized polygonal numbers. The notation $p_m$ and $\bar p_m$ is not defined in the supplied context, and no proof or resolution is given.