I am wondering if the following result is true:
Let $(V,\omega)$ be a symplectic vector space and $\{J_t\}_{0\leq t\leq 1}$ a smooth path of almost complex structures on $V$ which are tamed by $\omega$, i.e. $$J_t^2 = \mathbb{1}, \qquad\omega(v,J_t v) >0\quad \forall v \in V \backslash\{0\}.$$ Does there exists a path $\Phi_t \in GL(V)$ and a $\omega$-tame almost complex structure $\tilde{J}$ such that $$ \Phi_t^* J_t = \tilde{J}, \qquad \Phi_t^*\omega = \omega.$$
I would be thankfull for any answer and/or opinion wheather you expect this to be true or not.
Remark (Edit) If one replaces the tame condition by the condition of compatibility, the answer is yes. Any unitary trivialization with respect to the metric $g_t(\cdot,\cdot) := \omega(\cdot, J_t \cdot)$ would do the job. However, this argument seems not to carry over to the tamed case..