Multiple cover formula for parabolic stable-pair invariants on a curve

Let $C$ be the local curve considered in the paper, let $N_{n,\gamma}$ denote the corresponding generalized DT invariant for $(n,\gamma)\in\mathbb{Z}\oplus H_2(C,\mathbb{Z})$, and let $\mathrm{DT}^{\mathrm{par}}(\mu,C)$ be the parabolic stable-pair generating series in $\Lambda_C$. Multiple cover conjecture for the local curve. For $(n,\gamma)\in\mathbb{Z}\oplus H_2(C,\mathbb{Z})$,

$$N_{n,\gamma}=\sum_{k\geq 1,\,k\mid(n,\gamma)}\frac{1}{k^2}N_{1,\gamma/k}.$$

Equivalently, for every $\mu\in\mathbb{Q}$,

$$\mathrm{DT}^{\mathrm{par}}(\mu,C)=\prod_{\substack{\gamma\in H_2(C,\mathbb{Z})_{>0}\\ n/\omega\cdot\gamma=\mu}}\left(1-(-1)^{\gamma\cdot H}q^nt^{\gamma}\right)^{(\gamma\cdot H)N_{1,\gamma}}.$$

This is proposed as the local-curve analogue of the multiple cover and product-expansion conjectures. The source supplies no resolution, so the assertion remains open.