Prove that the following statement involving Buchstab function is true

Question

I came across this problem in this book called Cambridge studies in advanced mathematics 97, page 217-218. I wad reading this line where the author starts the inductive steps and apply the inductive hypothesis and yields $$\Phi(x,x^{\frac{1}{u}}) = \frac{U\omega(U)x}{\ln(x)}+O\bigg(\frac{x}{\ln^2(x)}\bigg)+\sum_{x^{\frac{1}{u}}\leq p<x^{\frac{1}{U}}} \bigg(\frac{u_p\omega(u_p)x}{p\ln(\frac{x}{p})}+O\bigg(\frac{p}{\ln(p)}\bigg)+O\bigg(\frac{x}{p\ln^2(x)}\bigg)\bigg)$$ which indeed is $$\Phi(x,x^{\frac{1}{u}}) = \frac{U\omega(U)x}{\ln(x)}+O\bigg(\frac{x}{\ln^2(x)}\bigg)+O\bigg(\frac{x^{\frac{2}{U}}}{\ln^2(x)}\bigg)+O\bigg(\frac{x}{\ln^2(x)}\bigg)+\sum_{x^{\frac{1}{u}}\leq p<x^{\frac{1}{U}}} \frac{u_p\omega(u_p)x}{p\ln(\frac{x}{p})}$$ since the author wrote that the sum over $p$ of the first error term is $\ll x^{\frac{2}{U}}/\ln^2(x)$, and the sum over $p$ of the second is $\ll x/\ln^2(x)$. The author then estimate the contribution of the main term in the sum by writing the Prime Number Theorem in the form $\pi(t)=\mathrm{li}(t)+R(t)$, then apply Riemann-Stieltjes integration and yields
$$\sum_{x^{\frac{1}{u}}\leq p<x^{\frac{1}{U}}} \frac{\omega(\frac{\ln(x)}{\ln(p)}-1)x}{p\ln(p)}=\int_{x^{\frac{1}{u}}}^{x^{\frac{1}{U}}} \frac{\omega(\frac{\ln(x)}{\ln(t)}-1)x}{t\ln^2(t)}\,dt+f(t)R(t)\bigg|_{x^{\frac{1}{u}}}^{x^{\frac{1}{U}}}-\int_{x^{\frac{1}{u}}}^{x^{\frac{1}{U}}} R(t)\,df(t)$$ where $$f(t)=\frac{\omega(\frac{\ln(x)}{\ln(t)}-1)x}{t\ln(t)}$$ which in fact it is $$\sum_{x^{\frac{1}{u}}\leq p<x^{\frac{1}{U}}} \frac{\omega(\frac{\ln(x)}{\ln(p)}-1)x}{p\ln(p)}=\int_{x^{\frac{1}{u}}}^{x^{\frac{1}{U}}} \frac{\omega(\frac{\ln(x)}{\ln(t)}-1)x}{t\ln^2(t)}\,dt+\int_{x^{\frac{1}{u}}}^{x^{\frac{1}{U}}} \frac{\omega(\frac{\ln(x)}{\ln(t)}-1)x}{t\ln(t)}\,dR(t) $$ before applying Riemann-Stieltjes integration. I noticed that for large $t$, we have $$dR(t)=\bigg(\frac{\ln(t)-1}{\ln^2(t)}-\frac{1}{\ln(t)}\bigg)\,dt=-\frac{dt}{\ln^2(t)}$$ I am also reading this paper on Dickman function and noticed the similar situation $$\sum_{x^d<p\leq x^{u_1}} \rho\bigg(\frac{\ln(p)}{\ln(x)-\ln(p)}\bigg)\frac{x}{p}=x\int_{x^d}^{x^{u_1}} \rho\bigg(\frac{\ln(t)}{\ln(x)-\ln(t)}\bigg)\,dF(t)$$ where $F(t)=\sum_{p\leq t} \frac{1}{p}$, therefore I guess that $\frac{1}{p\ln(p)}$ could also be expressed as such and my conjecture would be $$\sum_{x^{\frac{1}{u}}\leq p<x^{\frac{1}{U}}} \frac{\omega(\frac{\ln(x)}{\ln(p)}-1)x}{p\ln(p)}=\int_{x^{\frac{1}{u}}}^{x^{\frac{1}{U}}} \frac{(\ln(t)-1)\omega(\frac{\ln(x)}{\ln(t)}-1)x}{t\ln^3(t)}\,dt$$ The difficulty that I am facing is to prove the conjecture I made above, but currently I have no idea how this can be achieved. Is this conjecture correct and if it is correct, then how do I show it? Therefore I hope someone could explain the things that I am missing.