Modular Erdős distinct subset sums conjecture

Let $t\geq 0$ be fixed and let $n$ be sufficiently large. Set $N=2^n+t$, and let $A$ be an $n$-element set of positive integers whose subset sums are all distinct modulo $N$. Modular Erdős distinct subset sums conjecture. One has

$$\max A\geq \frac{N}{3}\geq \frac{2^n}{3}.$$

This is a modular version of the Erdős distinct subset sums problem, which asks for exponential lower bounds on the largest element of a set with distinct ordinary subset sums. The stated modular bound is presented as a conjecture; its resolution is not given in the supplied text.