Kileel's degree-independent bound conjecture for powers of polynomials

Let $d$, $k$, and $s$ be positive integers, and let $p_1,\ldots,p_k\in\mathbb{R}[x_1,\ldots,x_d]$ be homogeneous polynomials of the same degree $s$, no two of which are linearly dependent over $\mathbb{R}$. For sufficiently large $r$, the powers $p_1^r,\ldots,p_k^r$ are linearly independent; Proposition~ denotes a bound with this property by $\tilde{r}=\tilde{r}(d,k,s)$. Kileel's degree-independent bound conjecture. In the setting of Proposition~, $\tilde{r}$ may be taken to depend only on $d$ and $k$. This conjecture asks whether the threshold for linear independence can be chosen independently of the common degree $s$; the paper's abstract states that the conjecture is proved, with the resulting explicit bound given there.