Conjecture on defective weighted projective spaces

Let a weighted projective space be a projective toric variety, and call it defective when its dual variety is not a hypersurface. A cone over a weighted projective space is a weighted projective toric variety obtained as a cone over a lower-dimensional weighted projective space, with weights not necessarily reduced. Defectivity conjecture. The only defective weighted projective spaces are those which are cones over a weighted projective space of lower dimension, not necessarily with reduced weights. The paper reports computational evidence for weighted projective spaces of the form $\mathbb{P}(1,k,m,n)$ with $k,m,n\leq 10$, where the observed defective examples are precisely of the listed cone types; the general assertion remains open in the source.