The brane conjecture for open-closed BCOV theory and $6$-dimensional $(1,0)$ theories

Consider type IIA string theory on $\mathbb{R}^{10}$ with one $NS5$ brane on $\mathbb{R}^6\subset\mathbb{R}^{10}$, where $x_6,\ldots,x_9=0$, together with $k$ semi-infinite $D6$ branes on each side of the $NS5$ brane: one set on $x_6\geq 0$ and the other on $x_6\leq 0$, with $x_7,x_8,x_9=0$. The world-volume theory on the $NS5$ brane has $(1,0)$ supersymmetry. Brane conjecture. The holomorphic twist of this world-volume theory is open-closed BCOV theory whose open-sector gauge Lie algebra is $\mathfrak{gl}(k\mid k)$. This proposed relation connects the six-dimensional brane theory to the holomorphic twist studied in the paper; the surrounding discussion motivates it through the contributions of the two $D6$-brane collections and bifundamental strings, but does not establish it.