Non-vanishing stable instability conjecture for complex projective spaces

Let $k$ be a positive integer, and let $\operatorname{Conf}^n\mathbb{C}P^k$ denote the configuration space of $n$ ordered distinct points in complex projective $k$-space. Write $H^i(-;\mathbb{Q})$ for rational cohomology. The conjecture asserts that there are positive integers $n_0$, $c_s$, and $c_t$ such that, for

$$n \geq \frac{i-c_s}{2\lceil k/2\rceil-2},$$ $$H^i(\operatorname{Conf}^n\mathbb{C}P^k;\mathbb{Q})=H^i(\operatorname{Conf}^{n+1}\mathbb{C}P^k;\mathbb{Q}),$$

while for all $n\geq n_0$ and $i>(2\lceil k/2\rceil-2)n+c_s$,

$$H^i(\operatorname{Conf}^n\mathbb{C}P^k;\mathbb{Q})\approx H^{i+2\lceil k/2\rceil-2}(\operatorname{Conf}^{n+1}\mathbb{C}P^k;\mathbb{Q}).$$

Moreover,

$$H^{(2\lceil k/2\rceil-2)n+c_t}(\operatorname{Conf}^n\mathbb{C}P^k;\mathbb{Q})=H^{(2\lceil k/2\rceil-2)(n+1)+c_t}(\operatorname{Conf}^{n+1}\mathbb{C}P^k;\mathbb{Q})=\mathbb{Q}$$

for all $n\geq n_0$, and

$$H^{(2\lceil k/2\rceil-2)n+j}(\operatorname{Conf}^n\mathbb{C}P^k;\mathbb{Q})=H^{(2\lceil k/2\rceil-2)(n+1)+j}(\operatorname{Conf}^{n+1}\mathbb{C}P^k;\mathbb{Q})=0$$

for all $n\geq n_0$ and $j>c_t$. Non-vanishing stable instability conjecture. For every positive integer $k$, the stated stable range, shifted comparison, and top non-vanishing pattern hold. This extends the computed example for $\mathbb{C}P^3$ and is proposed as a general form of non-vanishing stable instability for configuration-space cohomology.