Arithmetic exponent pair hypothesis for smooth-conductor trace-function sums

Let $\rho=(0,1/2,0)$ be an arithmetic exponent pair. Given a sufficiently small $\eta>0$, let $q$ be a large, square-free, $q^{\eta}$-smooth number, let $K_q$ be an $\infty$-amiable trace function modulo $q$, and let $W_{\delta}$ be a $\delta$-periodic function. For an interval $I$, define

$$\mathfrak{S}(K,W;I)=\sum_{n\in I}K_q(n)W_{\delta}(n).$$

Arithmetic exponent pair hypothesis. For every $\epsilon>0$, provided that $|I|<q\delta$, one has

$$\mathfrak{S}(K,W;I)\ll_{\mathfrak{c},\epsilon,\eta}(q\delta)^{\epsilon}|I|^{1/2+\epsilon}.$$

This is a square-root cancellation hypothesis for sums involving smooth-conductor trace functions and periodic weights. The paper notes that its use still does not allow the method to break the $40\%$ barrier; the status of the hypothesis is not specified in the supplied text.