Optimal bounds for iterated sumsets with $|5A|=100$

Let $A$ be a subset of the integers, with iterated sumsets $hA=\{a_1+\cdots+a_h:a_i\in A\}$, and suppose that $|5A|=100$.

Optimal sumset bounds conjecture. Every such set satisfies

$$|4A|\ge 63\qquad\text{and}\qquad |6A|\le 145.$$

The example $A=\{0,1,5,8,49\}$ attains both bounds, so the conjecture asserts that the displayed estimates are optimal for finite sets of integers with $|5A|=100$. The source gives strong evidence but no proof or resolution.