Conjectural formula for the threshold at which two additional roots appear

Let $p\in\mathbb{Z}^+$, define $p^*=p\mod 9$, and let $n_0(p)$ denote the first value of $n$ for which the two additional solutions in the paper's theorem occur. Threshold formula conjecture.

$$n_0(p)=68\left\lfloor\frac{p}{9}\right\rfloor+7p^*+\left\lfloor\frac{p^*}{2}\right\rfloor.$$

The theorem establishes persistence of the additional roots after their appearance, while the displayed formula is presented as a numerical conjecture for the threshold. Its status is open in the supplied text.