No-common-edge shortest paths conjecture for multitriangulations

Let $T$ and $T'$ be $k$-triangulations of the $n$-gon, and let $G_{n,k}$ be their flip graph. A $k$-relevant edge is an edge eligible to occur in a $k$-triangulation.

No-common-edge shortest paths conjecture.

1. A shortest path in $G_{n,k}$ between $T$ and $T'$ never flips a common edge of $T$ and $T'$.
2. If $f$ is a $k$-relevant edge of $T$, $e$ is the unique bisector of the two $k$-stars of $T$ containing $f$, and $e$ is an edge of $T'$, then some shortest path from $T$ to $T'$ first flips $f$.

The source says the first assertion is known for ordinary triangulations and presents these statements as open in the multitriangulation setting.