KAM existence conjecture for whiskered tori in the Boussinesq system

Let $\omega^0$ be the vector of real linear frequencies of the Boussinesq system, and let $X=H^{\rho,m}(\mathbb T)\times H^{\rho,m+1}(\mathbb T)$. Fix a Diophantine exponent $\nu>\ell$ and a sufficiently large regularity exponent $m$. For a positive function $a$, write $B_{a(\varepsilon)}(\omega^0)\subset\mathbb R^\ell$ for the ball of radius $a(\varepsilon)$ centered at $\omega^0$.

KAM existence conjecture. There exist three explicit functions $a,b_d,b_a:\mathbb R^+\to\mathbb R^+$ such that

$$a(s)\to0,\qquad b_d(s),b_a(s)\to\infty\quad\text{as }s\to0,$$

and, for sufficiently small $\varepsilon$, every $\omega\in\mathcal D(b(\varepsilon),\nu)\cap B_{a(\varepsilon)}(\omega^0)$ admits an element $K\in X$ solving the invariant-torus equation for the parametrization of the Boussinesq water-wave system. Furthermore,

$$\frac{\left|D_h(b(\varepsilon),\nu)\cap B_{a(\varepsilon)}(\omega^0)\right|}{\left|B_{a(\varepsilon)}(\omega^0)\right|}\longrightarrow1.$$

The claim proposes the same type of large-measure KAM family for the Boussinesq water-wave system as for the preceding equation. The supplied text gives no resolution, while the surrounding discussion describes this as a result expected for related equations.