Equality of the extremal function and the first-order comparison function

Let $U(z)$ and $d6d1_1(z)$ be the functions defined in the paper for $z\in(0,2)$. Equality conjecture.

$$U(z)=\gamma_1(z)\quad\text{for }z\in(0,1].$$

This is one of the paper's open questions concerning the relationship between $U(z)$ and $\gamma_1(z)$; the source does not provide a resolution.