CPS rank decomposition conjecture for real nonsymmetric fourth-order tensors

Let $\mathcal{A}\in\mathbb{R}_{cps}^{n^4}\backslash\mathbb{R}_s^{n^4}$ be a real fourth-order conjugate partial-symmetric tensor that is not symmetric. A CPS rank decomposition is a decomposition of the form

$$\mathcal{A}=\sum_{i=1}^{R_1}\lambda_i(a_i^{\otimes2}\otimes\bar{a}_i^{\otimes2}+\bar{a}_i^{\otimes2}\otimes a_i^{\otimes2})+\sum_{j=1}^{R_2}\lambda_j b_j^{\otimes4},$$

where $\lambda_i,\lambda_j\in\mathbb{R}$, $a_i\in\mathbb{C}^n\backslash\mathbb{R}^n$, and $b_j\in\mathbb{R}^n$. CPS rank decomposition conjecture. Every such $\mathcal{A}$ has a CPS rank decomposition in this form. The preceding corollary establishes a CPS decomposition of this form, but the conjecture concerns its being a CPS rank decomposition; the supplied text gives no resolution.