Polynomial divisor-sum conjecture in short intervals

Let $f(x,y)$ be a fixed polynomial in $x$ and $y$ with integer coefficients, let $k$ be a positive integer, and let $N$ and $M$ satisfy

$$N^{\epsilon}<M<N^{1-\epsilon}.$$

Polynomial divisor-sum conjecture. For every $\epsilon>0$,

$$\sum_{N-M<n\le N}\tau_k(f(N,n))\ll M\log^{k-1}N$$

as $N\to\infty$. The conjecture combines results attributed in the paper to [E] and [S] and is used to seek sharper pointwise bounds for $R_3(n)$; the authors note that it may be known to experts but that they could not find a reference.