Tensoring by a plane maintains secant-regularity in degree at least two

Starting from an integral projective variety $Y$ equipped with a very ample, non-special and not-secant defective line bundle $\mathcal{L}$, the paper establishes, under certain conditions, the regularity of $(Y\times {\mathbb{P}}^2,\mathcal{L}[t])$ for $t\ge 2$. The mildness of those conditions allow to classify all secant defective cases of any product of $({\mathbb{P}}^1{)}^j\times ({\mathbb{P}}^2{)}^k$, $j,k\ge 0$, embedded in multidegree at least $(2,\dots ,2)$ and $({\mathbb{P}}^m\times {\mathbb{P}}^n\times ({\mathbb{P}}^2{)}^k,{\mathcal{O}}_{{\mathbb{P}}^m\times {\mathbb{P}}^n\times ({\mathbb{P}}^2{)}^k}(d,e,t_1,\dots ,t_k))$ where $d,e\ge 3$, $t_i\ge 2$, for any $n$ and $m$.