Conjecture on defective two-factor Segre-Veronese varieties

Let $\mathbf{n}=(m,n)\in\mathbb{N}^2$ and $\mathbf{a}=(a,b)\in\mathbb{N}^2$. Let $X_{\mathbf{n},\mathbf{a}}$ be the Segre-Veronese variety $\mathbb{P}^m\times\mathbb{P}^n$ embedded by $\mathcal{O}_{\mathbb{P}^m\times\mathbb{P}^n}(\mathbf{a})$. A pair $(\mathbf{n},\mathbf{a})$ is unbalanced when it satisfies the unbalancedness condition defined earlier in the paper. The variety $X_{\mathbf{n},\mathbf{a}}$ is defective when one of its secant varieties has smaller dimension than the expected dimension.

Defective Segre-Veronese conjecture. The variety $X_{\mathbf{n},\mathbf{a}}$ is defective if and only if $(\mathbf{n},\mathbf{a})$ falls into one of the following cases:

$$\begin{array}{ll} \text{(a)} & (\mathbf{n};\mathbf{a})=(m,n;a,1)\text{ is unbalanced and }m\geq2;\\ \text{(b)} & \mathbf{n}=(1,n)\text{ and }\mathbf{a}=(2k,2)\text{ with }k\geq1;\\ \text{(c)} & \mathbf{n}=(4,3)\text{, or }(2,n)\text{ with }n\text{ odd, and }\mathbf{a}=(1,2);\\ \text{(d)} & \mathbf{n}=(1,2)\text{ and }\mathbf{a}=(1,3);\\ \text{(e)} & \mathbf{n}=(2,2),(3,3)\text{, or }(3,4)\text{ and }\mathbf{a}=(2,2). \end{array}$$

The conjecture organizes the known defective cases of two-factor Segre-Veronese varieties; the paper reports evidence from results of several authors and from computation, while the asserted classification remains unproved in general.