Endpoint-limit conjecture for recurrence-defined functions

Let $\{F_n\}$, $\{G_n\}$ and $\{H_n\}$ be the sequences of real functions and let $g_n$, $h_n$, $g$, $h$ and $\Theta$ be as in the preceding pointwise convergence conjecture. Assume that

$$\Theta=\lim_{n\to\infty}\lim_{k\to\infty}F_n(n-k)$$

exists. Endpoint-limit conjecture. Then $F_n(\lfloor nx\rfloor)$ converges pointwise on $(0,1)$ to a function $f\in\mathcal{C}^1(0,1]$ satisfying

$$f'(x)=-g(x)+h(x)f(x),\qquad f(1)=\Theta.$$

This is a refinement of the preceding conjectural convergence principle, replacing the endpoint condition imposed through the limiting differential equation by an iterated limit of the recurrence values. The source provides no resolution, so the claim remains open.