Taylor-cancellation conjecture for sums of d-th roots

Let $d\geq 2$ and $k\geq 1$. Let $A_i$ be positive integers, let $u_i,v_i$ be integers, let $\lambda\ne 0$ be a real constant, and let $L(M)$ be an integer-valued linear polynomial in $M$. Taylor-cancellation conjecture. There exist such $A_i$, $u_i$, $v_i$, and $\lambda$ for which, for infinitely many integers $M$,

$$\sum_{i=1}^k A_i\left((M+u_i)^d+v_i\right)^{1/d}=L(M)+\lambda M^{-(dk-1)}+O_{d,k}(M^{-dk}).$$

Equivalently, after absorbing the weights $A_i$ into the radicands, one obtains $g_{k,d}(N)\ll_{d,k}N^{-(k-1/d)}$ for all sufficiently large $N$. This is presented as a natural Taylor-cancellation form of the integer-target conjecture; the supplied text gives no resolution, while the preceding theorem reaches only the weaker exponent $k/d-\varepsilon$.