Coarse Whitehead conjecture for spaces with finitely many ends

Let $X$ and $Y$ be compact metric spaces with finite shape dimension, let $\mathfrak{a}:cX\to cY$ be a coarse map, and suppose that $M_X$ has $t\in\mathbb{N}$ ends. Choose $[\tau_1],\ldots,[\tau_t]\in\pi_0^c(cX)$ whose images under $\theta$ represent these ends. Assume that, for every $\tau\in\{\tau_1,\ldots,\tau_t\}$ and every $n\geq0$, the induced map $\mathfrak{a}:\pi_n^c(cX,\tau)\to\pi_n^c(cY,\mathfrak{a}\tau)$ is an isomorphism. Finite-ended coarse Whitehead conjecture. There is an $l\in\mathbb{N}$ such that $\psi=(is_l\mathfrak{a}R)^\sim$ induces an isomorphism

$$b\circ\psi:\pi_n^e(M_X,\widetilde{i\tau})\to\pi_n^e(M_Y,\widetilde{is_l\mathfrak{a}\tau})$$

for all $n\geq0$ and all chosen $\tau$. If $\psi$ additionally induces an isomorphism on the ends of $M_X$ and $M_Y$, then $\mathfrak{a}$ is a coarse homotopy equivalence. This is proposed as an extension of coarse Whitehead from connected spaces to spaces whose model has finitely many ends.