Uniform exponent conjecture for sums of d-th roots

Let $d\geq 2$, $k\geq 1$, and $\varepsilon>0$. For $N\geq 1$ and $\beta\in\mathbb R/\mathbb Z$, consider the distance to the nearest integer, denoted by $\|\cdot\|$. Uniform exponent conjecture. For every fixed $d\geq 2$, $k\geq 1$, and $\varepsilon>0$,

$$\sup_{\beta\in\mathbb R/\mathbb Z} \min_{1\leq b_1,\ldots,b_k\leq N} \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\varepsilon} N^{-(k-1/d)+\varepsilon}.$$

The proved theorem gives only the exponent $k/d-\varepsilon$ using a sparse family of radicands, whereas this conjecture predicts the stronger exponent $k-1/d$ by exploiting the local freedom of general radicands. Its status is not specified in the supplied text.