The shifted-product nonempty-interior conjecture for product sets

Let $n\ge 3$ and let $A\subseteq \mathbb R$ be a Borel set. Write $\dim_H A$ for the Hausdorff dimension of $A$. Shifted-product nonempty-interior conjecture. For every $n\ge 3$, there exists $\varepsilon_n>0$ such that, if

$$\dim_H A>\frac{2}{n}-\varepsilon_n,$$

then there exist $t_1,\ldots,t_n\in\mathbb R$ such that

$$(t_1+A)(t_2+A)\cdots(t_n+A)$$

contains a nonempty open interval. This is expected to improve the direct threshold $\dim_H A>2/n$ for Euclidean shifted-product projections, but the conjectured improvement remains open.