Simple connectedness conjecture for evaluation spaces of bicategory objects

Let $t\in\Theta_2$ be an object, and let $h(t)\in\Theta_1$ denote its length. Let $\tau^-$ and $\tau^+$ be evaluations of $h(t)$, and let $E_2(t,\tau^-,\tau^+)$ be the corresponding evaluation space for bicategory objects in 2-categories. Simple connectedness conjecture. The evaluation space $E_2(t,\tau^-,\tau^+)$ is simply connected, for all objects $t\in\Theta_2$ and all evaluations $\tau^-$ and $\tau^+$ of the length $h(t)\in\Theta_1$. This would ensure that the definition of a bicategory object in a 2-category is complete, by showing that all relations among the specified 1-cell modifications are generated by the stated 2-cells.