Frieze–Lovász generic rank conjecture for three-way tensors

Let $m,n,p$ be integers in the critical range specified by the source, and define

$$r_0(m,n,p)=\left\lceil\frac{mnp}{m+n+p-2}\right\rceil.$$

Let $r_{\mathrm{gen}}(m,n,p)$ denote the generic rank of tensors in $\mathbb{C}^{m\times n\times p}$.

Frieze–Lovász generic rank conjecture. Equality

$$r_{\mathrm{gen}}(m,n,p)=r_0(m,n,p)$$

holds unless $(m,n,p)=(3,2k+1,2k+1)$ for $k\in\mathbb{N}$. In this exceptional case,

$$r_{\mathrm{gen}}(3,2k+1,2k+1)=r_0(3,2k+1,2k+1)+1.$$

This is a conjectural formula for the generic rank in the critical range. The exceptional case is stated in the source as already known, while the general assertion is attributed there to the cited literature; no resolution of the remaining cases is supplied in the provided text.