On non-secant defectivity of Segre-Veronese varieties

Let SV^{\pmb n}_{\pmb d} be the Segre-Veronese variety given as the image of \mathbb{P}^{n_1}\times \dots \times \mathbb{P}^{n_r} under the embedding induced by the complete linear system \big \vert\mathcal {O}_{\mathbb{P}^{n_1}\times \dots \times \mathbb{P}^{n_r}}(d_1,\dots , d_r)\big \vert. We prove that asymptotically SV^{\pmb n}_{\pmb d} is not h-defective for h\leq (\min \{n_i\})^{\lfloor \log _2(d-1)\rfloor }, where d = d_1+\dots +d_r.