Continuous orthonormal trivialization conjecture for continuous projector families

Let $G$ be the underlying group and let $cmathbb{K}^d$ have canonical basis $cxi_1,\dots,\xi_d$. Let $\mathrm{P}:G\to \mathbb{B}_X$ be a continuous projector. For every $g\in G$, choose an orthonormal basis $\{\eta_1^g,\dots,\eta_n^g\}$ of $\operatorname{im}\mathrm{P}(g)$ and an orthonormal basis $\{\eta_{n+1}^g,\dots,\eta_d^g\}$ of $\ker\mathrm{P}(g)$.

Continuous orthonormal trivialization conjecture. The bases can be chosen so that the map $T:G\to GL_d(\mathbb{K})$ defined by

$$T(g)\xi_i=\eta_i^g$$

is continuous.

This conjecture asks whether every continuous family of projectors admits a continuous choice of orthonormal bases for its image and kernel, equivalently a continuous trivialization adapted to the projector. The supplied text gives no evidence resolving the question.