The extension conjecture for complex shifted powers

Let $d$ be a positive integer, and let

$$F=\{(x-a_i)^{e_i}:1\leq i\leq s\}$$

be a linearly independent family with $a_i\in\mathbb{C}$, $e_i\leq d$ for all $i$, and $s\leq ad+b$. Extension conjecture. There are absolute constants $a$ and $b$ such that the family

$$F\cup\{(x+1)^{d+1},x^{d+1}\}$$

is linearly independent as well. This is a strengthening related to the large-exponent conjecture: it asks when two prescribed powers of degree $d+1$ can be adjoined without creating a dependence. Its resolution is not supplied in the source context.