Natural density of thickly syndetic set.

Question

A syndetic set $S$ is a subset of the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded. That is to say, $S$ is a syndetic set, if there exist a positive integer $l$, such that for any $n$ , we have $\{n,n+1,⋯,n+l\} \cap S$ is non-empty. Set T is thickly syndetic if $$\bigcap_{i=1}^{N} T + i $$ is syndetic for each $N$, where $T + i = \{t + i \;| \; \forall t \in T\}$. Is it true, that natural density $d(T) = 1?$ i.e. $$\lim_{n \rightarrow \infty} \frac{|T \cap \{1, ...,n\} |}{n} = 1 ?$$