The triple convolution conjecture for generalised divisor functions

For integers $k,l,m\ge 2$, define

$$\mathcal{T}(d_k,d_l,d_m;x,h):=\sum_{n\le x}d_k(n+h)d_l(n)d_m(n-h).$$

For $\epsilon>0$ and $0<h\le x^{1-\epsilon}$, the triple convolution conjecture.

$$\mathcal{T}(d_k,d_l,d_m;x,h)\sim \nabla_{h,k,l,m}\frac{x(\log x)^{k+l+m-3}}{(k-1)!(l-1)!(m-1)!},$$

as $x\to\infty$, where $\nabla_{h,k,l,m}$ is the explicit Euler-product constant specified in the source, with $g(u,v,w)=1$ when the congruence system $n\equiv-h\pmod u$, $n\equiv0\pmod v$, $n\equiv h\pmod w$ has a solution and $g(u,v,w)=0$ otherwise. This generalises Browning's open conjecture for $d(n+h)d(n)d(n-h)$; the paper establishes a lower bound of the correct order, but the asymptotic remains unproved.