Pointwise convergence conjecture for recurrence-defined functions

Let $\{F_n\}$, $\{G_n\}$ and $\{H_n\}$ be sequences of real functions on $\{1,\ldots,n\}$ satisfying

$$F_n(k)=G_n(k)+H_n(k)F_n(k+1),\qquad F_n(n)=\mu.$$

Define $g_n(x):=nG_n(\lfloor nx\rfloor)$ and $h_n(x):=n(1-H_n(\lfloor nx\rfloor))$ on $[0,1]$, and suppose that they converge pointwise on $(0,1)$ to continuous functions $g$ and $h$. Suppose also that the differential equation

$$y'(x)=-g(x)+h(x)y(x)$$

admits a solution on $(0,1]$ only for the final condition $y(1)=\Theta$. Pointwise convergence conjecture. Then $F_n(\lfloor nx\rfloor)$ converges pointwise on $(0,1)$ to a function $f\in\mathcal{C}^1(0,1]$ satisfying

$$f'(x)=-g(x)+h(x)f(x)\quad\text{for }x\in(0,1),\qquad f(1)=\Theta.$$

The conjecture seeks sufficient conditions for pointwise, rather than uniform, convergence of functions associated with recurrence relations when the limiting differential equation determines the endpoint value. The source provides no resolution, so the claim remains open.