Lurie's dualizability conjecture for $E_n$-algebras

Let $n$ be a nonnegative integer, let $\mathcal{S}$ denote the ambient symmetric monoidal category of spaces, and let $\mathcal{A}\in\operatorname{Alg}_n(\mathcal{S})$ be an $E_n$-algebra. For each $k=0,\ldots,n$, write $\int_{S^{k-1}\times\mathbb{R}^{n-k+1}}\mathcal{A}$ for its factorization homology, with the specified framing, acting on $\mathcal{A}\cong\int_{\mathbb{R}^n}\mathcal{A}$. Lurie's dualizability conjecture. The algebra $\mathcal{A}$ is $(n+1)$-dualizable if, and only if, it is dualizable over the factorization homologies

$$\int_{S^{k-1}\times\mathbb{R}^{n-k+1}}\mathcal{A}$$

for $k=0,\ldots,n$. The forward implication is clear; the case $n=1$ is proved by Lurie, and the case $n=2$ is proved in the source, while the general characterization remains conjectural.