Given $\lambda>1$ and $d\in\mathbb N$, we consider the set $$\mathcal W(\lambda,d)=\{\sum_{0\leq x_1\leq x_2\leq \dots\leq x_d}x_1^\lambda+\ldots+x_d^\lambda\}$$ of all possible sums of (at most) $d$ integers to the power $\lambda$.
Question: Does there always exist a smallest integer $w(\lambda)$ such that $\mathcal W(\lambda,w(\lambda))$ has only bounded gaps?
If yes and if $\lambda\not\in\mathbb N$ does there exist an integer $w'(\lambda)\geq w(\lambda)$ such that $\mathcal W(\lambda,w'(\lambda))$ has only a finite number of gaps of size at least $\epsilon$ for any strictly positive $\epsilon$?
Easy observation: $w(\lambda)$ exists for all $\lambda>1$ in $\mathbb Q$ by the original proof for Waring numbers (for $\lambda=a/b$ consider only integers of the form $n^b$). Similarly, $w'(\lambda)$ exists for all $\lambda\in\mathbb Q\setminus\mathbb N$ larger than $1$.
Trivial observation on the original Waring problem: The definition of $w(\lambda)$ is also of limited interest for $\lambda$ integral: We have for example $w(2)=3$. Indeed, for any $k$ there are $k$ consecutive integers which are not sums of two squares: Consider $N+1,\ldots,N+k$ such that $N+j\equiv p_{i_j}\pmod{p_{i_j}^2}$ for $k$ distinct primes $p_{i_1},\ldots, p_{i_k}$ in $3+4\mathbb N$. On the other hand, the set $4^n(7+8m)$ of all integers which are not sums of three squares contains no consecutive elements. (The definition of $w'$ is of course meaningless for $\lambda\in\mathbb N$.)
Final remark: The question concerning $w(\lambda)$ is essentially equivalent to a question on integers: Round $x^\lambda$ to some integer at distance at most $B$ (with $B$ arbitrary but independent of $x$). If the set of all rounded integers is coprime, almost all natural integers are sums of a bounded number (given by a suitable increase of $w(\lambda)$) of such rounded powers.