On the representation of friable integers by linear forms

Let $P^+(n)$ denote the largest prime of the integer $n$. Using the \begin{align*}\Psi\_{F\_1\cdots F\_t}\left(\mathcal{K}\cap[-N,N]^d,N^{1/u}\right):=\\#\left\{\mathcal{K}\in {\mathbf{N}}\cap[-N,N]^d:\vphantom{P^+(F\_1(\boldsymbol{n})\cdots F\_t(\boldsymbol{n}))\leq N^{1/u}}\right.\left.P^+(F\_1(\boldsymbol{n})\cdots F\_t(\boldsymbol{n}))\leq N^{1/u}\right\}\end{align*} where $(F\_1,\ldots,F\_t)$ is a system of affine-linear forms of $\mathbf{Z}[X\_1,\ldots,X\_d]$ no two of which are affinely related and $\mathcal{K}$ is a convex body. This improves upon Balog, Blomer, Dartyge and Tenenbaum's work~\cite{BBDT12} in the case of product of linear forms.