Rigidity conjecture for the families $(\varphi_\alpha\Phi_n)_{\alpha\in U}$

Let $U$ be an open set of $\mathbb{C}^d$, let $n\geq 3$ be an integer, and let $\varphi_\alpha$ be a holomorphic family of matrices in $\mathrm{PGL}(3;\mathbb{C})$ parameterized by $U$. Assume that there exists a positive integer $k$ such that

$$(k+1)(2n-1)\geq 10,$$

for $0\leq i\leq k-1$,

$$(\varphi_\alpha\Phi_n)^i\varphi_\alpha(P)\neq P,$$

and

$$(\varphi_\alpha\Phi_n)^k\varphi_\alpha(\widehat{\xi}_2)=\widehat{\xi}_1.$$

Rigidity conjecture. Then the family $(\varphi_\alpha\Phi_n)_{\alpha\in U}$ is holomorphically trivial.