Is it true that if $A=\{a_1<\cdots <a_t\}\subseteq \{1,\ldots,N\}$ has no solutions to $$a_i+a_{i+1}+\cdots+a_j\in A$$ then $$\lvert A\rvert \leq \frac{N}{2}+O(1)?$$

The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the maximal size of such an $A$ satisfies $$\lvert A\rvert \leq \left(\frac{2}{3}-\frac{1}{512}\right)N+\log N.$$