The weak Whitney theorems for harmonic surfaces

Let $M$ be the target manifold and let $\operatorname{\mathfrak{M}}$ denote the moduli space of harmonic surfaces in $M$. A harmonic immersion is a harmonic map that is an immersion, and a harmonic embedding is a harmonic map that is an embedding.

Weak Whitney theorems. The space of harmonic immersions in $\operatorname{\mathfrak{M}}$ is open and dense, and connected if $\dim M\geq 5$; moreover, the space of harmonic embeddings in $\operatorname{\mathfrak{M}}$ is open and dense, and connected if $\dim M\geq 6$.

The preceding local results establish these properties near somewhere injective harmonic maps with isolated singularities, while this conjecture asserts them globally without the isolated-singularity hypothesis. It predicts generic immersion and embedding behavior for harmonic surfaces in the indicated target dimensions.