The polynomial zero-avoidance conjecture on the unit ball

Let $P_1,\ldots, P_N\in \mathbb R[x_1, \ldots, x_d]$ be polynomials, and let $\delta_1,\ldots, \delta_N>0$ satisfy

$$\sum_{k=1}^N \delta_k\deg P_k\leq 1.$$

Polynomial zero-avoidance conjecture. There exists a point $p\in B^d\subset\mathbb R^d$ such that, for every $k=1,\ldots,N$, the point $p$ is at distance at least $\delta_k$ from the zero set of $P_k$.

This conjecture simultaneously generalizes Bang's theorem and zero-avoidance results for polynomials. The supplied text gives partial results and a complex projective analogue, but does not state that the real-ball conjecture has been resolved.