We study the homotopy theory of positive loops in the real symplectic group $\mathrm{Sp}(2n)$. We prove that two positive loops are homotopic if and only if they are homotopic through positive loops, thereby providing a positive answer to a question by Lalonde and McDuff in 1997. As geometric applications, we remove the four–dimensional restrictions in results of McDuff and Chance on Hamiltonian loops with fixed global maxima: their uniruledness criteria extend verbatim to arbitrary dimension. In particular, if a nontrivial element of $\pi_1(\mathrm{Ham}(M,\omega))$ admits a Hofer length–minimizing representative, then $(M,\omega)$ is uniruled. This is a joint work with Qinglong Zhou.