The partially symmetric rank of a product of two W-states and of bihomogeneous binary monomials

For $d_1,d_2\ge 2$ let $W_d=x^{d-1}y\in\mathrm{Sym}^d(\mathbb C^2)$ (up to scale) be the W-state. We prove that the partially symmetric rank of $W_{d_1}\otimes W_{d_2}$ equals $2(d_1+d_2-2)$, so the known upper bound of Gałązka and of Canino–Casarotti–Santarsiero is sharp for every pair. This is the two-factor case of Question 12 of Oneto–Ventura. The lower bound combines two elementary facts: a point set whose Veronese images span $x^py\otimes u^qv$ lies on no curve of bidegree $(1,q)$ or $(p,1)$, and a Sylvester-type catalecticant inequality, in the form used by Wang–Seigal, evaluated at this unbalanced pair of bidegrees. The same argument gives $R(x^ay^b\otimes u^cv^d)=(a+1)(c+1)-(a-b)(c-d)$ for all $a\ge b\ge1$, $c\ge d\ge1$, and a lower bound for products of binary forms whose rank exceeds their border rank. The case of three or more factors is not settled.