Defectivity conjecture for Taylor varieties

Let $\mathcal{T}^n_{d,e,m}$ denote the Taylor variety, and call it defective when its dimension is smaller than the expected dimension. Defectivity conjecture. The following statements hold: (1) for $n=2$, all Taylor varieties $\mathcal{T}^2_{d,e,m}$ are non-defective; (2) for $n=3$, there are only seven defective Taylor varieties, namely those listed in the source's table; and (3) for fixed $n\geq3$, only finitely many triples $(d,e,m)$ yield a defective $\mathcal{T}^n_{d,e,m}$. The conjecture is based on computational experiments and extends the observed finite lists of defective examples; the general assertions are not proved in the supplied text.