Exact value of the zero-sum invariant for rank-two groups

Prove that the zero-sum invariant \(\mathsf N(C_p\oplus C_{np})\) equals \(2np-1\) for every integer \(n\ge2\) and every prime \(p\ge3\) that is the smallest prime divisor of \(np\).

Background

The paper establishes the lower bound $\mathsf N(C_p\oplus C_{np})\ge 2np-1$ and proves equality when $p\mid n$. When $p\nmid n$, it obtains the upper bounds $5np-4$ for $p\ge5$ and $4np-3$ for $p=3$. The author observes that the unresolved cases would follow from the bound $c_0(C_p\oplus C_{np})\le\lceil(4np-2)/3\rceil$, via Proposition 5.1.