Independence of the Bott index from the polynomial or logarithmic map

Let $U_1$ and $U_2$ be approximately commuting unitary matrices, with $obreak\bigl\Vert[U_1,U_2]\bigr\Vert\leq ecdelta$, and let $H_1,H_2,H_3$ be the associated Hermitian matrices constructed using functions $f,g,h$. For sufficiently small $ecdelta$, consider the index $\widetilde{\operatorname{bott}}(H_1,H_2,H_3)$. Map-independence conjecture. The index $\widetilde{\operatorname{bott}}(H_1,H_2,H_3)$ is the same whether $f,g,h$ are defined by the polynomial map or by the logarithmic map. The surrounding text gives a deformation argument intended to establish this claim and refers to the resulting spectral gap and constant eigenvalue counts; however, the supplied candidate is labeled as a conjecture and no external resolution status is provided.