Uniform boundedness conjecture for Waring's problem over diagonal forms

Let $k\in\mathbb{N}$. For each positive integer-valued function $C:\mathbb{N}\rightarrow\mathbb{N}$, let $P_C(k)=\sup_{l\geq 2}\{S_C(k,l)\}$, where $S_C(k,l)$ is the minimum number of variables $s$ for which the relevant equation is soluble over the set $\mathcal{T}_{C(k)l}$ by the Hardy–Littlewood method. Uniform boundedness conjecture. There exists a positive integer-valued function $C:\mathbb{N}\rightarrow\mathbb{N}$ such that

$$P_C(k)<\infty.$$

The conjecture asserts that, for a suitable choice of $C$, the number of variables required can be bounded independently of $l$ once $k$ is fixed. The paper explains that the available lower bound for $\lvert\mathcal{T}_t\cap[1,N]\rvert$ is insufficient to prove this statement and establishes a weaker result instead.