The conjecture that the modified Erdős–Ginzburg–Ziv constant of $Z_p^3$ equals $9p-8$

Let $p$ be a prime, and let $s_1(Z_p^3)$ denote the least integer $\ell$ such that every sequence of length $\ell$ in the finite abelian group $Z_p^3$ contains a zero-sum subsequence of length $p$. Conjecture.

$$s_1(Z_p^3)=9p-8.$$

The conjecture gives the expected exact value of this zero-sum constant; the surrounding discussion notes that proving a lower bound or an upper bound of comparable size would be significant. Its status is not resolved in the supplied source.