Equality of partial theta series for tensor products involving projective modules

Let $p$ be a non-negative integer, and let

$$F_i=M_0^{\oplus\alpha_i}\oplus M_{-2}^{\oplus\beta_i}\oplus P^{\oplus\gamma_i}$$

for $0\leq i\leq p$. Define

$$F'_i=M_0^{\oplus(\alpha_i+\gamma_i)}\oplus M_{-2}^{\oplus(\beta_i+\gamma_i)}.$$

Partial theta series conjecture. The partial theta series associated with the tensor products satisfy

$$\chi\left(q,l,\bigotimes_{i=0}^{p}F_i\right)=\chi\left(q,l,\bigotimes_{i=0}^{p}F'_i\right).$$

The claim says that the contributions of the summands $P$ can be replaced by equal additional multiplicities of $M_0$ and $M_{-2}$ without changing the partial theta series. The supplied text gives no evidence about whether this equality has been proved or remains open.