Small-perturbation solvability conjecture for the Cantor series system

Let $t\in\mathbb{N}_3$ and let $\vec{\epsilon}=(\epsilon_1,\ldots,\epsilon_t)\in\mathbb{R}^t$. Consider the nonlinear system

$$\begin{aligned} c_0+c_1+c_2+c_3+\cdots+c_{t-1}&=2t+\epsilon_1\\ c_0c_1+c_1c_2+c_2c_3+c_3c_4+\cdots+c_{t-2}c_{t-1}&=2t+\epsilon_2\\ c_0c_1c_2+c_1c_2c_3+c_2c_3c_4+c_3c_4c_5+\cdots+c_{t-3}c_{t-2}c_{t-1}&=2t+\epsilon_3\\ \cdots\\ c_0c_1\cdots c_{t-1}&=2t+\epsilon_t. \end{aligned}$$

Small-perturbation solvability conjecture. Whenever $\lVert\vec{\epsilon}\rVert$ is sufficiently small, this system has a solution

$$(c_0,c_1,\ldots,c_{t-1})\in[t,t+1]\times\left[1+\frac{1}{2t},1+\frac{1}{t-1}\right]^{t-1}.$$

The source notes computer verification for $t\leq100$ when $\vec{\epsilon}=0$ and explains that this conjecture would imply the AP-abnormality conjecture and further order-separation results. It remains unproved.