The weak-integrability conjectures for Alexandrov submetries

Let $f:X\to Y$ be a submetry with $X\in \operatorname{Alex}^n(\kappa)$ and $Y\in \operatorname{Alex}^m(\kappa)$. A submetry is weakly integrable when the local isometric pieces in the source's definition exist, and integrable when in addition their spaces of directions equal the horizontal spaces.

Weak-integrability conjectures. (1) If $\partial Y=\emptyset$ and $f$ is weakly integrable, then $f$ is integrable. (2) If $\Sigma\in \operatorname{Alex}^n(1)$, $\Sigma_0\in \operatorname{Alex}^m(1)$, $\partial\Sigma_0=\emptyset$, and $h:\Sigma\to\Sigma_0$ is weakly integrable, then $n=m$.

The source states that these two assertions are equivalent and that they fail when the boundary restrictions are removed. It also notes that the first is immediate for Riemannian manifolds, while the second remains nontrivial there.