Can an Isomap be embedded in a manifold of higher dimension than the corresponding MDS?

I am using the Isomap algorithm to operate a dimension reduction on a distance matrix $M_{dist}$. For a given choice of nearest neighbors k to compute the geodesic distance, I use the following method to determine the dimensionality of the corresponding manifold: I compute a distance matrix of D-simensional gaussian vectors, with the same average square distance as $M_{dist}$, which I call $M_{rand}$. I then use a loop to compute the reconstruction error $e_{dist}$ of the Isomap fitted on $M_{dist}$ and the reconstruction error $e_{rand}$ of the Isomap fitted on $M_{rand}$ for the whole range of possible number of components for the Isomap, so for $d\in[1,dim(M_{dist})]$. I then consider the number of components maximizing the ratio $\frac{e_{rand}}{e_{dist}}$ as the intrinsic dimensionality of the manifold.

Using this method I have computed the intrinsic dimensionality of the Isomap obtained for every possible number of nearest neighbors $k\in[1,dim(M_{dist})-1]$. I obtain the following graph: Isomap intrinsic dimension depending on number of nearest neighbors which increases from 4 to about 25 and then decreases to 14 when $k=dim(M_{dist}-1)$. This borderline case actually corresponds to a Multi-Dimensional Scaling (MDS) algorithm.

Therefore, using an MDS algorithm, my data seems to be comprised in a 14-dimension space. Does it make sense to consider an Isomap projection with a parameter k (number of nearest neighbors) resulting in an embedding of higher dimension than 14?