Conjecture on uniform high-degree bounds for powers of homogeneous polynomials

Let $d$, $k$, and $s$ be integers, and let $p_1,\ldots,p_k\in\mathbb{R}[x_1,\ldots,x_d]$ be homogeneous polynomials of the same degree $s$, with no two linearly dependent. Proposition~ asserts that there is a threshold $\tilde r=\tilde r(d,k,s)$ such that $p_1^r,\ldots,p_k^r$ are linearly independent whenever $r>\tilde r$. Uniform high-degree bound conjecture. In this setting, $\tilde r$ may be taken to depend only on $d$ and $k$. This conjecture would make the high-activation-degree linear-independence threshold uniform in the common polynomial degree $s$; no resolution is supplied in the given text.