Sparsity conjecture for sets satisfying the prime number theorem

For real $x$ and $t>1$, let $S_{\left\lfloor x/n^t\right\rfloor,x}$ denote the set considered in the paper, with the function defining $S_{f,x}$ taken to be non-piecewise. Sparsity conjecture. The family of sets $S_{\left\lfloor x/n^t\right\rfloor,x}$ are the sparsest sets that satisfy the prime number theorem. The preceding asymptotic formula shows that these sets satisfy the prime number theorem, while their density becomes smaller as $t$ increases; the conjecture asserts optimality of this sparsity among such sets.