Simple-current fusion rules for singlet-algebra modules

Let $p\in\mathbb{Z}_{\ge 2}$. For $j\in\mathbb{Z}$, let $\pi_j$ be the irreducible $\overline{M(1)}$-module defined in the surrounding construction, and let $M_t$ denote the corresponding ordinary, $\mathbb{N}$-graded $\overline{M(1)}$-module. Simple-current fusion conjecture. The modules $\pi_j$ are simple currents in the category of ordinary, $\mathbb{N}$-graded $\overline{M(1)}$-modules, with fusion rules

$$\pi_j\times M_t=M_{t-jp}.$$

This predicts that tensoring with each $\pi_j$ permutes the irreducible ordinary $\overline{M(1)}$-modules according to the displayed rule. The supplied source does not state whether this claim has been proved or disproved.