Towards a version of Markov's theorem for ribbon torus-links in $\mathbb{R}^4$

In classical knot theory, Markov's theorem gives a way of describing all braids with isotopic closures as links in $\mathbb{R}^3$. We present a version of Markov's theorem for extended loop braids with closure in $B^3 \times S^1$, as a first step towards a Markov's theorem for extended loop braids and ribbon torus-links in $\mathbb{R}^4$.