Conjecture on the growth of AP_k-covering sequences

Let $\mathbb{N}$ denote the positive integers, and let $A\subseteq\mathbb{N}$. For an integer $k\geq 3$, call $A$ an $AP_k$-covering sequence if there exists an integer $n_0$ such that, for every $n>n_0$, there are elements $a_1,\ldots,a_{k-1}\in A$ with

$$a_1<\cdots<a_{k-1}<n$$

for which $a_1,\ldots,a_{k-1},n$ form a $k$-term arithmetic progression. Let

$$A(n)=\sum_{\substack{a\in A\a\leq n}}1$$

be the number of elements of $A$ up to $n$.

Growth conjecture for $AP_k$-covering sequences. (i) For every integer $k\geq 3$, there exists an $AP_k$-covering sequence $A$ such that

$$\limsup_{n\rightarrow\infty}\frac{A(n)}{n^{\frac{k-2}{k-1}}}<\infty.$$

(ii) For every $AP_k$-covering sequence $A$, there is a constant $c_k>0$, with $k\geq 3$, such that

$$\liminf_{n\rightarrow\infty}\frac{A(n)}{n^{\frac{k-2}{k-1}}}>c_k.$$

The preceding results establish constructions with an additional logarithmic factor in the upper bound and give the lower bound $\liminf_{n\to\infty}A(n)/\sqrt n\geq\sqrt2$ when $k=3$; the conjecture asks whether the logarithmic factor can be removed and whether the power $n^{(k-2)/(k-1)}$ is the correct general order of growth.