Equivalence of left, partitionable and tensor-action convergence for uniform hypergraphs

Let $H_n=(V(H_n),E(H_n))$ be a sequence of $r$-uniform hypergraphs, and let $A(H_n)$ denote the normalized adjacency tensors. Left-convergence is convergence of homomorphism densities, and partitionable convergence is the hypergraph convergence notion defined through branching partitions.

Convergence-equivalence conjecture. Left-convergence, partitionable convergence, and action convergence of the $(r-1)$-action of the sequence

$$\frac{A(H_n)}{|V(H_n)|}$$

of normalized adjacency tensors are equivalent.

Partitionable convergence is already known in the source to imply left-convergence, while the conjecture asserts equivalence with both left-convergence and the $(r-1)$-action convergence of normalized adjacency tensors.